Numerical Simulation of Primary Atomization of Non-Newtonian Impinging … · 2013-04-10 · ILASS Americas, 25th Annual Conference on Liquid Atomization and Spray Systems, Pittsburgh,

ILASS Americas 25th Annual Conference on Liquid Atomization and Spray Systems Pittsburgh PA May 2013

Numerical Simulation of Primary Atomization of Non-Newtonian Impinging Jets

Xiaodong Chen and Vigor Yang School of Aerospace Engineering Georgia Institute of Technology

270 Ferst Drive NW Atlanta GA 30332 USA

Abstract Newtonian impinging jets have been investigated for decades However limited is known about non-Newtonian impinging jets The present work attempts to improve this situation by performing high-fidelity numerical simula-tions of non-Newtonian impinging jet dynamics Emphasis is placed on the near field of the liquid sheet formed by the impingement of two shear-thinning (viscosity deceases with increasing shear rate) liquid jets and the ensuing atomization The formulation is based on a complete set of conservation equations for both the liquid and surround-ing gas phases An improved volume-of-fluid (VOF) method combined with an innovative topology-oriented adap-tive mesh refinement (TOAMR) technique is developed and implemented to track the interfacial dynamics The stringent requirements for grid resolution for fine and thin structures are resolved efficiently in the computational domain Especially the refinement at the interface to the minimum required grid size is treated effectively to sub-stantially reduce the simulation cost Results show good agreement with available experimental data in terms of sheet topology and breakup length of the unstable hydrodynamics wave known as the impact wave Comparison is made with Newtonian liquid jets to identify the underlying physical mechanisms of jet dynamics sheet formation impact waves and atomization The work allows for investigation into the evolution of the shear rates in the liquid sheet and at the interface It is found that the viscosity decreases in the region where the impingement of the two liquid jets results in a high shear rate Increasing the volume of this low viscosity region can enhance atomization Information obtained from the present study can be used to optimize the design parameters of non-Newtonian im-pinging jet injectors including the impingement angle jet diameter and off-center distance of the two jets

Introduction Collision of two cylindrical jets is one of the ca-

nonical configurations for the generation of liquid sheets The dynamics and stability of liquid sheets have attracted a great deal of attention due to their relevance to the atomization and combustion processes in liquid rocket engines[1-4] Impingement is an efficient meth-od for atomizing and mixing the liquid jets The dynam-ic head of one of the jets is used to destabilize the op-posing jet stream thereby resulting in fragmentation of the jet into ligaments and droplets [5] The atomization of impinging jets of Newtonian fluids has been exten-sively investigated experimentally and theoretically Dombrowski and Hooper [6] investigated the factors that affect the breakup of sheets and studied wave mo-tions (impact wave) of high velocity liquid sheets An-derson et al[7] developed a phenomenological three-step model of atomization including atomization of the impinging jets and sheet and ligament breakup pro-cesses provided a good correlation with the experi-mental data Theoretical studies have been limited due to the excessive complexity of relevant physical pro-cesses Preliminary works focused on atomization of the impinging jets [8-11] Recently Chen et al[12] performed high-fidelity numerical simulations and in-vestigated phenomena responsible for the formation of the impact wave The Strouhal number locking-on fea-ture of an impact wave was found to be similar to that observed for perturbed free shear layers thereby indi-cating that the flow mechanism of an impact wave is analogous to that of free shear layers[13]

Gelled propellants feature many advantages over non-gelled liquid and solid rocket propellants They are safe to store and handle and comply with insensitive munitions requirements The inherent thrust modulation capability provides good application flexibility for utili-zation in tactical rocket engine Gelled propellants are non-Newtonian fluids since their viscosity depends on the shear rate They are difficult to atomize and require higher injection pressures As a result it would be ad-vantageous for the gelled propellants to be shear-thinning liquids whose viscosity decreases with increas-ing shear rate The formation and breakup mechanisms of non-Newtonian liquid sheets formed by impinging jets are poorly known Fakhri [14] studied the devel-opment and atomization process of non-Newtonian impinging jets of gelled propellant simulants The phys-ical and rheological properties of gelled simulants are identical to those of conventional hypergolic propel-lants Near-field spray characteristics such as the sheet formation and breakup length of the liquid sheet were determined The droplet size distributions were meas-ured in the far field region downstream of the im-pingement location Coil [15] studied the effect of the gel phase on the atomization of impinging jets The spray behaviors of Newtonian oil and gelled oil were

compared using high-speed images Yang [16] per-formed a linear instability analysis and predicted the breakup length and wavelength for a non-Newtonian liquid exhibiting power-law behavior The results were in good agreement with the experimental data

Direct numerical simulation (DNS) can be used to determine the effect of shear-thinning on the atomiza-tion process of non-Newtonian liquids The result can be used to guide the injector design by improving the performance and reducing the required pressure of the supply system The present study focuses on analyzing the atomization process of non-Newtonian impinging jets Emphasis is placed on physical processes that oc-cur in the vicinity of the impingement location Theoretical Framework

a Numerical methods Atomization is a complex physical process featuring

large density ratios and significant viscous and surface tension forces It involves dynamic and complex inter-facial geometries with length scales spanning several orders of magnitude An open-source flow solver GERRIS[17] is employed in the present study The nu-merical method [18] combines an adaptive quadOctree spatial discretisation geometrical Volume-Of-Fluid interface representation balanced-force continuum-surface-force surface tension formulation and height-function curvature estimation It recovers exact equilib-rium (to machine accuracy) between surface tension and pressure gradient for the case of a stationary droplet irrespective of the viscosity and spatial resolution The three-dimensional incompressible variable-density conservative equations can be written as

( ) 0tρ ρpart +nabla sdot =u (1)

( ) (2 )t spρ micro σκδpart + sdot nabla = minusnabla + nabla sdot +u u u D n (2)

0nabla sdot =u (3)

where u is the fluid velocity ρ is the fluid density μ is the dynamic viscosity σ the surface tension coefficient D the deformation tensor and κ and n denote the curva-ture and normal to the interface respectively The Dirac delta function δs is used to include the surface tension term only at the interface

A volume-of-fluid (VOF) function c=c(xt) is calcu-lated to track the multi-fluid interface It is defined as the volume fraction of a given fluid in each cell of the computational grid The density and viscosity are de-fined as

1 2( ) (1 )c c cρ ρ ρequiv + minus (4)

1 2( ) (1 )c c cmicro micro microequiv + minus (5)

where w the subscripts 1 and 2 denote the first and second fluid respectively Field c is either equal to c or is constructed by applying a smoothing spatial filter to c For high liquid-gas density ratios more accurate results are obtained if a smoothed field is employed to calculate the viscosity When spatial filtering is used field c is calculated by averaging the eight corner val-ues of c which are obtained using bilinear interpolation of the cell-centered values The properties associated with the interface are thus ldquosmearedrdquo over three dis-cretization cells[17 18]

The advection equation for the volume fraction is given by

( ) 0tc cpart + nabla sdot =u (6)

A temporally staggered discretization of the vol-ume-fraction or density and pressure leads to a second-order accuracy in time[17] This system is further sim-plified using a classical time-splitting projection meth-od [19] which requires the solution of a Poisson equa-tion The standard multi-grid scheme exhibits slow convergence for elliptic equations with discontinuous coefficients andor source terms especially for flows with large-density-ratios [18] In order to improve the performance the discretized momentum equation is rearranged into Helmholtz-type equation which can be solved using a variant of the multilevel Poisson solver The Crank-Nicholson discretization of the viscous terms is second-order accurate and unconditionally sta-ble This numerical scheme is stable for CFL numbers lower than one

Spatial discretization is performed using a graded Octree partitioning All the variables are collocated at the center of the cubic discretization volume and are interpreted as volume-averaged values which is con-sistent with the finite volume formulation The collocat-ed definition facilitates conservation of momentum when dealing with adaptive meshes [17]

To solve the advection equation for the volume frac-tion Popinet [17] employed a piecewise-linear geomet-rical Volume-of-Fluid (VOF) scheme generalized for the QuadOctree spatial discretization The interface is represented in each cell by a line (resp plane in three dimensions) which is described by the equation mx = α where m is the local normal to the interface and x is the position vector Given m and the local volume frac-tion c α is uniquely determined by ensuring that the volume of fluid contained in the cell and lying below the plane is equal to c This volume can be computed relatively easily by taking into account the different ways a square (resp cubic) cell can be cut by a line (resp plane) which leads to matched linear and quadrat-ic (resp cubic) functions of α

The accurate estimation of the surface tension term in the discretized momentum equation is one of the issues concerning the application of VOF methods to surface-tension-driven flows[17] The original Contin-uum-Surface-Force (CSF) [20] is known to suffer from problematic parasitic currents for the case of a station-ary droplet in equilibrium [21] Other methods based on phase-field description of the interface suffer from similar problems including level-sets and front tracking with distributed surface tension Popinet [17] showed that the combination of a balanced-force surface tension discretization and a Height-Function curvature estima-tion resolves the issue of parasitic currents provided sufficient time is given for the initial non-equilibrium interface shape to relax to its equilibrium configuration Relaxation occurs on a timescale comparable to that of the viscous dissipation process The non-equilibrium shape converges numerically towards the equilibrium configuration at second-order rate

The Height-Function technique is relatively easy to extend to an Octree spatial discretization It however may be inconsistent for the radius of curvature of the interface lower than approximately five times the grid spacing In such cases the parabolic fitting technique is used The transition between the techniques has been shown to be consistent with overall second-order accu-racy

The overall scheme allows for the resolution to vary spatially and temporally To simplify the implementa-tion the variation of the size of neighboring cells was restricted to a factor of two (this is sometimes referred to as restricted Octree) This may pose some adaption issues for three-dimensional flows that have fractal di-mension close to two Fortunately this is not an issue for most complex fluid dynamics problems In contrast to previous implementations this method is not limited to constant resolution along the interface This can dramatically increase the efficiency of mesh adaptation particularly when dealing with reconnections and breakup of interfaces One of the advantages of the Oc-tree discretization is that mesh refinement or coarsening are cheap and can be performed at every time-step with minimal impact on the overall performance Interpola-tion of quantities on newly refined or coarsened cells is relatively simple and is done conservatively for mo-mentum and volume fraction [17] The refinement level of the root cell is zero The level of its children cells is one and so on recursively A cell of level n has a resolu-tion of 2n in each coordinate

b Simulation setup In the present study a like doublet impinging jet

configuration is employed It features two streams of the same liquid It is the common type of injector con-figuration Figure 1 shows the schematic of the low speed impingement process A thin liquid sheet is

formed at the center of two liquid jets The angle be-tween the axis of the jets 2α is the impingement angle The two important non-dimensional parameters are Weber number ρuj

2djσ and Reynolds number ρujdjμ where ρ is the liquid density dj the jet diameter uj the mean jet velocity σ is the surface tension and μ is vis-cosity of liquid

Figure 1 Schematic diagram of doublet impinging jets

The rheology and flow characteristics of non-Newtonian fluids are different from those of Newtonian fluids Newtonian fluids have constant viscosities while the viscosities of non-Newtonian fluids vary with applied shear rate The composition of the working flu-id was adopted from Ref [14] It consists of TS-720 silica (5 wt ) and 981A Carbopol (01 wt ) in 7525 by vol ethanolwater mixture This simulant gel is used in order to match rheological and physical prop-erties of typical Gelled Hypergolic Propellants (GHPs)[22] The results of numerical simulations are compared with the experimental data [14] The relation-ship between shear rate and viscosity can be represent-ed accurately using the extended Herschel-Bulkley model [23]

n-10 Kτ

micro γ microγ infin= + +

where μ is non-Newtonian viscosity τ0 is yield stress γ is shear rate K is pre-exponential coefficient n is pow-er law index and μinfin is non-Newtonian viscosity at infi-nite shear Figure 2 shows rheological property of the gel employed in this work The Weber number of the simulant gel takes the same form as that of a Newtonian

liquid A modified formula for Reynolds number is used since the viscosity of the gel changes with shear rate[24 25]

Figure 2 Rheology of simulant gel

c Refinement criteria Since adaptive mesh refinement (AMR) uses multi-

ple-levels mesh care must be taken to ensure the accu-racy of the results In present method several refine-ment criteria are used depending on the problem For interfacial flows four basic refinement criteria are ap-plied to improve the accuracy and efficiency There are gradient-based value-based curvature-based and to-pology-orientated refinements

For interfacial flows it is important to ensure that the interface is adequately resolved Gradient-based refinement is used so that a fine grid is present only at the interface The kinetic energy in the liquid phase is yet another important parameter for consideration A fine mesh is required to avoid excessive dissipation of kinetic energy due to numerical viscosity The interior of the specific phase can be refined according to the volume fraction of the fluid The lengths scale changes dramatically across the computational domain during primary atomization In order to resolve the topology changes fine grid is necessary for small diameters and thicknesses For example it is expensive to refine the interface grid based on the minimum diameter since the curvature of interface changes temporally and spa-tially The curvature-based refinement can be used to resolve the fine structures Since the thickness of the thin structure is difficult to obtain an experimental re-finement method adapted from digital topologic theo-ry[26] is extended and implemented to ensure that there are at least certain number of grids in the thickness di-rection We call this method Topology-Oriented Adap-tive Mesh Refinement (TOAMR)

Figure 3(a) is the overall structure of the simulation result Both thickness- and curvature- based refine-ments are used to ensure the accuracy The fish-bone

2α inlet jets

rear point

tip pointθ=π

θ=π2

dj=2rj

Shear Rate (1s)

100 101 102 103

like flow structure is caught The size scale and thick-ness vary in a wide range from jet diameter to thinning liquid sheet It is clear that with the thickness-based method the level cell changes automatically across the simulation domain Figure 3(b) shows a detail interfa-cial cell near the impact point From the impact point to the downstream of the liquid film the cell level in-creases according to the decreased film thickness Fig-ure 3(c) shows the cell in the cross section It is clear that with both thickness and location changing of the liquid sheet the topology-oriented refinement method always ensures that there are at least two cells in the thickness direction

(a) (b)

Figure 3 Mesh and interface of 3-D impinging jets case (a) overall interface with cell edges (b) close view at impact point with cell edges (c) mesh and in-terface on cross section

Results and Discussion a Flow patterns Figure 4 compares the near field transients of liquid

sheet formation with experimental images[14] at three different inlet velocities (corresponding Weber numbers for the three cases from top to bottom are 1549 6195 and 12390 respectively) Good qualitative agreements are obtained in terms of the flow patterns and sheet topology at steady state The flow patterns obtained in numerical simulation appear to be more stable than the

experimental ones For example with We=1549 a flat liquid sheet is obtained in the simulation in contrast to irregular ripples noticed in the experimental images This difference is caused by the inlet turbulence in the experiments The shapes of liquid jets are disturbed as shown clearly in the experimental images It must be noted that turbulence of the inlet liquid jets is not taken into account in the present model Adaptive mesh re-finement during the simulation provides highly resolved images of the flow structures and atomization process which will be discussed in detail in the following sec-tions

Figure 4 Comparison of flow patterns under different Weber numbers (Images in the left row are obtained from experiments[14] and in the right row depict pre-sent simulation results The Weber number for the three cases from top to bottom are 1549 6195 and 12390 respectively)

b Rim instability Figure 5 consists of a series of images showing the

development of flow pattern at We=1549 A flat liquid sheet bounded by the rim is formed at the center when the two liquid sheets come in contact As the size of the liquid sheet increases an asymmetrical wave appears as shown in Figure 5(c) This wave is of the same type as the cardioids observed by Taylor[27] As the liquid sheet continuously expands it ruptures near the rear point as shown in Figure 5(d) After that the liquid rim at the rear point is disturbed The disturbances propa-gate and develop along the two sides of the liquid sheet The boundary of the liquid sheet ruptures resulting in the shedding of the rim to form ligaments as shown in Figure 5(f) to (l) The ligaments break into droplets

under the effect of surface tension Due to fluid viscosi-ty the shedding of the liquid rim results in web-like structures and the droplets are less visible as shown in Figure 5(k) The steady state of this process is shown in Figure 5(l) It is interesting to notice the small ampli-

tude surface waves on the liquid sheet The damped amplitude is caused by the high viscosity of the simu-lant gel The structure of these surface waves at higher We will be discussed in the next section

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

Figure 5 Development of flow pattern under We=1549 (interface colored by y coordinate)

c Impact wave Figure 6 shows a series of images describing the

development of flow pattern at We=6195 In this case the velocity of the liquid rim is high enough to cause capillary instabilities which lead the formation of liga-ments and droplets near the boundary of the liquid sheet When the two liquid jets fully impact each other as shown in Figure 6(d) wave structures appear from the

rear point The wave propagates downstream with the liquid sheet as shown in Figure 6(e) to (i) The overall wave pattern is similar to previous simulation results[13] of a Newtonian fluid This wave is usually called ldquoim-pact waverdquo The thin regions of liquid sheet are torn off by the impact wave The gaps contract into liquid liga-ments distributed by the two sides of the liquid sheet

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 6 Development of flow pattern at We=6195 (interface colored by y coordinate)

A similar case depicting the propagation of impact waves at We = 12390 is shown in Figure 7 Since the inlet velocity is higher the droplet shedding at the be-ginning of the impingement process result in formation of a droplet cloud (see Figure 7(a) to (d)) The ampli-tude of the impact waves are higher compared to the previous case with We=6195 As the wave propagate

downstream the thickness of liquid sheet decreases dramatically The liquid sheet fully ruptures at about four to five wavelengths from the impact point This observation agrees well with the experimental images[14] The ligaments under this higher inlet veloc-ity are longer and thinner as shown in Figure 8

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

Figure 8 Comparison of detail atomization process at We=6159 (left) and We=12390 (right)

Strouhal number is defined as St=fDUs=Dλ where f is wave propagation frequency D the jet diameter Us the velocity of liquid sheet and λ is the wavelength The values of St under different We are plotted in Fig-ure 9 It is clear that the Strouhal numbers are almost the same for the three cases The simulations carried on in this paper are in the range of locking-on Previous study shows that the Strouhal number is locked behind We=1000 for Newtonian liquid Present results indicts that the critical Weber number may be the same for all the liquid Also the locking-on value of the Strouhal number for this non-Newtonian liquid is similar with previous study This indicates that the wavelength is independent of viscosity and depends only on geomet-rical parameters ie jet diameter and impingement an-gle

Figure 9 Locking-on of Strouhal number

d Shear-thinning effect Figure 10 shows the distributions of shear rate at

cross section for Weber numbers 1549 6195 and 12390 respectively The dark red regions indicate large shear rates Figure 10(a) shows that large shear rates are

found near the boundaries and near the impact point of the two jets The former can be attributed to the veloci-ty difference at the inlet orifice whereas the latter is due to high velocity gradient For the cases with impact waves (see Figure 10(b) and (c)) the low viscosity re-gions are much larger than that of stable liquid sheet It is also interesting to notice that the distributions of shear rate show wave-like structures which propagate downstream Careful observation shows that these wave-like structures are formed due to oscillation of liquid sheet at the rear point Another evidence for this conclusion is the propagation of the captured gas bub-bles in Figure 10(c) The reduction in gel viscosity indi-cates that with the same initial viscosity the non-Newtonian liquid is easier to atomize than the viscous Newtonian fluid for the impinging jets configuration

Figure 10 Distribution of shear rate at the cross sec-tion under different Weber number (a) We=1549 (b) We=6195 (c) We=12390

Figure 11 shows the distribution of shear rate at the interfaces for the three cases For We=1549 high shear rate is is found on the liquid sheet near the rear point and also on the liquid rim When We increases to 6195 the high shear rate region near the rear point widens The shear rates on the wave crest and trough are higher than at other location on the waves Since the ligaments are formed by the ruptured liquid sheet the shear rate on the ligaments are higher than that in inlet jets This is helps in better atomization The shear rates in the thin part of the liquid ligaments are highest as shown in Figure 11(b) It is caused by the stretching of liquid

5000 1000004

ligaments which results in higher shear rate and lower viscosity This is a well-known phenomenon for a shear-thinning non-Newtonian liquid which is called ldquobeads on a stringrdquo describing a form of spherical drop-lets connected by thin ligaments Figure 11(c) shows more complicated breakup process of liquid ligaments with higher inlet velocity There are more twisted liga-ments long ligaments with dramatically changed diam-eters and finer droplets

Figure 11 Distribution of shear rate on the interface under different Weber number (a) We=1549 (b) We=6195 (c) We=12390

Conclusions and Future Work Numerical simulations are performed to study the

characteristics of liquid sheets formed by two non-Newtonian impinging jets The efficient numerical methods used in present paper allow high fidelity re-

sults of near-field fluid dynamics Good overall agree-ments are obtained with available experiments The detailed flow-field of the impact waves are captured for the first time by numerical simulations The dynamics of atomization process of non-Newtonian liquid is compared with Newtonian liquid The result from pre-vious study on impinging jets of Newtonian liquid is extended to the non-Newtonian shear-thinning liquid It is also found that viscosity of shear-thinning liquid sheet can be reduced by the impinging jets configura-tion The optimization of the design parameters for example impingement angle jet diameter and off-center distance of the two jets can improve the atomi-zation characteristics to fit the requirement of breakup length and droplet size distribution In the near future simulations with a larger computational domain will be carried out to explore the far-field dynamics of non-Newtonian impinging jets

Acknowledgments This work was sponsored by the US Army Re-

search Office under the Multi-University Research Ini-tiative with contract No W911NF-08-1-0124 The sup-port and encouragement provided by Dr Ralph An-thenien are gratefully acknowledged Special thanks is due to Dr Steacutephane Popinet for allowing us to use his VOF and AMR algorithms

References 1 Bayvel L and Z Orzechowski Liquid atomization Vol 1040 1993 CRC 2 Lefebvre AH Atomization and sprays 1989 CRC 3 Sutton GP and O Biblarz Rocket propulsion ele-ments 2011 Wiley 4 Bush JWM and AE Hasha On the collision of laminar jets fluid chains and fishbones Journal of Flu-id Mechanics 2004 511 p 285-310 5 Anderson WE HM Ryan and RJ Santoro Im-pinging jet injector atomization Liquid rocket engine combustion instability(A 96-11301 01-20) Washington DC American Institute of Aeronautics and Astro-nautics Inc(Progress in Astronautics and Aeronautics 1995 169 p 215-246 6 Dombrowski N and P Hooper A study of the sprays formed by impinging jets in laminar and turbu-lent flow J Fluid Mech 1963 18(3) p 392-400 7 Anderson WE HM Ryan III and RJ Santoro Impact Wave-Based Model of Impinging Jet Atomiza-tion Atomization and Sprays 2006 16(7) p 791-806 8 Inoue C T Watanabe and T Himeno Study on Atomization Process of Liquid Sheet Formed by Im-pinging Jets in 44th AIAAASMESAEASEE Joint Propulsion Conference amp Exhibit 2008 American In-stitute of Aeronautics and Astronautics 9 Inoue C T Watanabe and T Himeno Liquid Sheet Dynamics and Primary Breakup Characteristics at Im-

pingement Type Injector in 45th AI-AAASMESAEASEE Joint Propulsion Conference amp Exhibit 2009 American Institute of Aeronautics and Astronautics 10 Arienti M et al Coupled Level-SetVolume-of-Fluid Method for the Simulation of Liquid Atomization in Propulsion Device Injectors in 46th AI-AAASMESAEASEE Joint Propulsion Conference amp Exhibit 2010 American Institute of Aeronautics and Astronautics 11 Li X et al Towards an Efficient High-Fidelity Methodology for Liquid Jet Atomization Computations in 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition 2010 American Institute of Aeronautics and Astro-nautics 12 Chen X D Ma and V Yang Mechanism Study of Impact Wave in Impinging Jets Atomization in 50th AIAA Aerospace Sciences Meeting AIAA Nashville Tennessee 2012 13 Chen X D Ma and V Yang High-Fidelity Nu-merical Simulations of Impinging Jet Atomization in 48th AIAAASMESAEASEE Joint Propulsion Con-ference amp Exhibit Atlanta Georgia 2012 14 Fakhri SAK A Study on the Atomization and Spray Characteristics of Gelled Simulants Formed by Two Impinging Jets 2009 The Pennsylvania State University 15 Coil M Effect of Viscoelastic Properties in Im-pinging Jet Sprays in ICLASS 2009 ICLASS 16 Yang L-j et al Breakup of a power-law liquid sheet formed by an impinging jet injector International Journal of Multiphase Flow 2011 17 Popinet S Gerris a tree-based adaptive solver for the incompressible Euler equations in complex geome-tries Journal of Computational Physics 2003 190(2) p 572-600 18 Popinet S An accurate adaptive solver for surface-tension-driven interfacial flows Journal of Computa-tional Physics 2009 228(16) p 5838-5866 19 Chorin A On the convergence of discrete approx-imations to the Navier-Stokes equations Mathematics of Computation 1969 23(106) p 341-353 20 Brackbill J D Kothe and C Zemach A continu-um method for modeling surface tension 1 Journal of Computational Physics 1992 100(2) p 335-354 21 Popinet S and S Zaleski A front-tracking algo-rithm for accurate representation of surface tension International Journal for Numerical Methods in Fluids 1999 30(6) p 775-793 22 Rahimi S et al Preparation and characterization of gel propellants and simulants AIAA paper 2001(2001-3264) 23 Natan B and S Rahimi The status of gel propel-lants in year 2000 Combustion of energetic materials 2001 p 172-194

24 Metzner A and J Reed Flow of non‐newtonian fluidsmdashcorrelation of the laminar transition and turbu-lent‐flow regions AIChE Journal 2004 1(4) p 434-440 25 Dodge D and A Metzner Turbulent flow of non‐newtonian systems AIChE Journal 2004 5(2) p 189-204 26 Tchon KF et al Three-dimensional anisotropic geometric metrics based on local domain curvature and thickness Computer-Aided Design 2005 37(2) p 173-187 27 Taylor G Formation of thin flat sheets of water Proceedings of the Royal Society of London Series A Mathematical and Physical Sciences 1960 259(1296) p 1-17

Xiaodong Chen and Vigor Yang

School of Aerospace Engineering

Georgia Institute of Technology


Abstract

Introduction

Theoretical Framework

a Numerical methods

b Simulation setup

c Refinement criteria

Results and Discussion

a Flow patterns

b Rim instability

c Impact wave

d Shear-thinning effect

Conclusions and Future Work

Acknowledgments

References

Introduction Collision of two cylindrical jets is one of the ca-

nonical configurations for the generation of liquid sheets The dynamics and stability of liquid sheets have attracted a great deal of attention due to their relevance to the atomization and combustion processes in liquid rocket engines[1-4] Impingement is an efficient meth-od for atomizing and mixing the liquid jets The dynam-ic head of one of the jets is used to destabilize the op-posing jet stream thereby resulting in fragmentation of the jet into ligaments and droplets [5] The atomization of impinging jets of Newtonian fluids has been exten-sively investigated experimentally and theoretically Dombrowski and Hooper [6] investigated the factors that affect the breakup of sheets and studied wave mo-tions (impact wave) of high velocity liquid sheets An-derson et al[7] developed a phenomenological three-step model of atomization including atomization of the impinging jets and sheet and ligament breakup pro-cesses provided a good correlation with the experi-mental data Theoretical studies have been limited due to the excessive complexity of relevant physical pro-cesses Preliminary works focused on atomization of the impinging jets [8-11] Recently Chen et al[12] performed high-fidelity numerical simulations and in-vestigated phenomena responsible for the formation of the impact wave The Strouhal number locking-on fea-ture of an impact wave was found to be similar to that observed for perturbed free shear layers thereby indi-cating that the flow mechanism of an impact wave is analogous to that of free shear layers[13]

Gelled propellants feature many advantages over non-gelled liquid and solid rocket propellants They are safe to store and handle and comply with insensitive munitions requirements The inherent thrust modulation capability provides good application flexibility for utili-zation in tactical rocket engine Gelled propellants are non-Newtonian fluids since their viscosity depends on the shear rate They are difficult to atomize and require higher injection pressures As a result it would be ad-vantageous for the gelled propellants to be shear-thinning liquids whose viscosity decreases with increas-ing shear rate The formation and breakup mechanisms of non-Newtonian liquid sheets formed by impinging jets are poorly known Fakhri [14] studied the devel-opment and atomization process of non-Newtonian impinging jets of gelled propellant simulants The phys-ical and rheological properties of gelled simulants are identical to those of conventional hypergolic propel-lants Near-field spray characteristics such as the sheet formation and breakup length of the liquid sheet were determined The droplet size distributions were meas-ured in the far field region downstream of the im-pingement location Coil [15] studied the effect of the gel phase on the atomization of impinging jets The spray behaviors of Newtonian oil and gelled oil were

compared using high-speed images Yang [16] per-formed a linear instability analysis and predicted the breakup length and wavelength for a non-Newtonian liquid exhibiting power-law behavior The results were in good agreement with the experimental data

Direct numerical simulation (DNS) can be used to determine the effect of shear-thinning on the atomiza-tion process of non-Newtonian liquids The result can be used to guide the injector design by improving the performance and reducing the required pressure of the supply system The present study focuses on analyzing the atomization process of non-Newtonian impinging jets Emphasis is placed on physical processes that oc-cur in the vicinity of the impingement location Theoretical Framework

a Numerical methods Atomization is a complex physical process featuring

large density ratios and significant viscous and surface tension forces It involves dynamic and complex inter-facial geometries with length scales spanning several orders of magnitude An open-source flow solver GERRIS[17] is employed in the present study The nu-merical method [18] combines an adaptive quadOctree spatial discretisation geometrical Volume-Of-Fluid interface representation balanced-force continuum-surface-force surface tension formulation and height-function curvature estimation It recovers exact equilib-rium (to machine accuracy) between surface tension and pressure gradient for the case of a stationary droplet irrespective of the viscosity and spatial resolution The three-dimensional incompressible variable-density conservative equations can be written as

( ) 0tρ ρpart +nabla sdot =u (1)

( ) (2 )t spρ micro σκδpart + sdot nabla = minusnabla + nabla sdot +u u u D n (2)

0nabla sdot =u (3)

where u is the fluid velocity ρ is the fluid density μ is the dynamic viscosity σ the surface tension coefficient D the deformation tensor and κ and n denote the curva-ture and normal to the interface respectively The Dirac delta function δs is used to include the surface tension term only at the interface

A volume-of-fluid (VOF) function c=c(xt) is calcu-lated to track the multi-fluid interface It is defined as the volume fraction of a given fluid in each cell of the computational grid The density and viscosity are de-fined as

1 2( ) (1 )c c cρ ρ ρequiv + minus (4)

1 2( ) (1 )c c cmicro micro microequiv + minus (5)

n-10 Kτ

2α inlet jets

rear point

tip pointθ=π

θ=π2

dj=2rj

Shear Rate (1s)

100 101 102 103

(a) (b)

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

5000 1000004





Abstract

Introduction


a Numerical methods

b Simulation setup



a Flow patterns

b Rim instability

c Impact wave



Acknowledgments

References

n-10 Kτ

2α inlet jets

rear point

tip pointθ=π

θ=π2

dj=2rj

Shear Rate (1s)

100 101 102 103

(a) (b)

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

5000 1000004





Abstract

Introduction


a Numerical methods

b Simulation setup



a Flow patterns

b Rim instability

c Impact wave



Acknowledgments

References

n-10 Kτ

2α inlet jets

rear point

tip pointθ=π

θ=π2

dj=2rj

Shear Rate (1s)

100 101 102 103

(a) (b)

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

5000 1000004





Abstract

Introduction


a Numerical methods

b Simulation setup



a Flow patterns

b Rim instability

c Impact wave



Acknowledgments

References

(a) (b)

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

5000 1000004





Abstract

Introduction


a Numerical methods

b Simulation setup



a Flow patterns

b Rim instability

c Impact wave



Acknowledgments

References

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

5000 1000004





Abstract

Introduction


a Numerical methods

b Simulation setup



a Flow patterns

b Rim instability

c Impact wave



Acknowledgments

References

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

5000 1000004





Abstract

Introduction


a Numerical methods

b Simulation setup



a Flow patterns

b Rim instability

c Impact wave



Acknowledgments

References

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

5000 1000004





Abstract

Introduction


a Numerical methods

b Simulation setup



a Flow patterns

b Rim instability

c Impact wave



Acknowledgments

References

5000 1000004





Abstract

Introduction


a Numerical methods

b Simulation setup



a Flow patterns

b Rim instability

c Impact wave



Acknowledgments

References

Documents

Numerical Simulation of Primary Atomization of Non-Newtonian Impinging … · 2013-04-10 · ILASS Americas, 25th Annual Conference on Liquid Atomization and Spray Systems, Pittsburgh,