Numerical Simulation of Primary Atomization of Non-Newtonian Impinging Jets

Transcription

1 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 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 simulations 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 surrounding gas phases. An improved volume-of-fluid (VOF) method, combined with an innovative topology-oriented adaptive 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 substantially 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 impinging jet injectors, including the impingement angle, jet diameter, and off-center distance of the two jets.

2 Introduction Collision of two cylindrical jets is one of the canonical 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 method for atomizing and mixing the liquid jets. The dynamic head of one of the jets is used to destabilize the opposing jet stream, thereby resulting in fragmentation of the jet into ligaments and droplets [5]. The atomization of impinging jets of Newtonian fluids has been extensively investigated experimentally and theoretically. Dombrowski and Hooper [6] investigated the factors that affect the breakup of sheets and studied wave motions (impact wave) of high velocity liquid sheets. Anderson et. al[7] developed a phenomenological threestep model of atomization including atomization of the impinging jets, and sheet and ligament breakup processes., provided a good correlation with the experimental data. Theoretical studies have been limited due to the excessive complexity of relevant physical processes. Preliminary works focused on atomization of the impinging jets [8-11]. Recently, Chen et. al[12] performed high-fidelity numerical simulations and investigated phenomena responsible for the formation of the impact wave. The Strouhal number locking-on feature of an impact wave was found to be similar to that observed for perturbed free shear layers, thereby indicating 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 utilization 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 advantageous for the gelled propellants to be shearthinning liquids whose viscosity decreases with increasing shear rate. The formation and breakup mechanisms of non-newtonian liquid sheets formed by impinging jets are poorly known. Fakhri [14] studied the development and atomization process of non-newtonian impinging jets of gelled propellant simulants. The physical and rheological properties of gelled simulants are identical to those of conventional hypergolic propellants. Near-field spray characteristics such as the sheet formation and breakup length of the liquid sheet were determined. The droplet size distributions were measured in the far field region downstream of the impingement 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] performed 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 atomization 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 occur 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 interfacial geometries with length scales spanning several orders of magnitude. An open-source flow solver, GERRIS[17] is employed in the present study. The numerical method [18] combines an adaptive quad/octree spatial discretisation, geometrical Volume-Of-Fluid interface representation, balanced-force continuumsurface-force surface tension formulation, and heightfunction curvature estimation. It recovers exact equilibrium (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 + ( ρu ) = 0 (1) t ρ ρ ( u+ u u) = p + (2 µ D) + σκδ n (2) t u =0 (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 curvature and normal to the interface, respectively. The Dirac delta function δ s is used to include the surface tension term only at the interface; A volume-of-fluid (VOF) function c=c(x,t) is calculated 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 defined as ρ( c ) c ρ + (1 c ) ρ 1 2 µ ( c ) c µ + (1 c ) µ 1 2 s (4) (5)

3 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 values of c, which are obtained using bilinear interpolation of the cell-centered values. The properties associated with the interface are, thus, smeared over three discretization cells[17, 18]. The advection equation for the volume fraction is given by c+ ( cu ) = 0 t (6) A temporally staggered discretization of the volume-fraction or density and pressure leads to a secondorder accuracy in time[17]. This system is further simplified using a classical time-splitting projection method [19], which requires the solution of a Poisson equation. The standard multi-grid scheme exhibits slow convergence for elliptic equations with discontinuous coefficients and/or 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 stable. 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 consistent with the finite volume formulation The collocated definition facilitates conservation of momentum when dealing with adaptive meshes [17]. To solve the advection equation for the volume fraction, Popinet [17] employed a piecewise-linear geometrical Volume-of-Fluid (VOF) scheme generalized for the Quad/Octree spatial discretization. The interface is represented in each cell by a line (resp. plane in three dimensions), which is described by the equation m x = α, where m is the local normal to the interface and x is the position vector. Given m and the local volume fraction 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 quadratic (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 Continuum-Surface-Force (CSF) [20] is known to suffer from problematic parasitic currents for the case of a stationary 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 estimation 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 accuracy. The overall scheme allows for the resolution to vary spatially and temporally. To simplify the implementation, 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 dimension 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 Octree discretization is that mesh refinement or coarsening are cheap and can be performed at every time-step with minimal impact on the overall performance. Interpolation of quantities on newly refined or coarsened cells is relatively simple and is done conservatively for momentum 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 resolution of 2 n 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 configuration. Figure 1 shows the schematic of the low speed impingement process. A thin liquid sheet is

4 formed at the center of two liquid jets. The angle between the axis of the jets, 2α is the impingement angle. The two important non-dimensional parameters are Weber number, ρu j 2 d j /σ, and Reynolds number, ρu j d j /μ, where ρ is the liquid density, d j the jet diameter, u j the mean jet velocity, σ is the surface tension, and μ is viscosity of liquid. liquid. A modified formula for Reynolds number is used, since the viscosity of the gel changes with shear rate,[24, 25] rear point u j 2α θ=0 y z u j inlet jets d j =2r j x θ=π/2 Viscosity (Pa.s) 10 0 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 fluid was adopted from Ref. [14]. It consists of TS-720 silica (5 wt. %) and 981A Carbopol (0.1 wt. %) in 75/25 by vol. ethanol/water mixture. This simulant gel is used in order to match rheological and physical properties of typical Gelled Hypergolic Propellants (GHPs)[22]. The results of numerical simulations are compared with the experimental data [14]. The relationship between shear rate and viscosity can be represented accurately using the extended Herschel-Bulkley model [23],: τ γ tip point 0 n-1 µ = + Kγ + µ θ=π (7) where μ is non-newtonian viscosity, τ 0 is yield stress, γ is shear rate, K is pre-exponential coefficient, n is power law index, and μ is non-newtonian viscosity at infinite 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 r Shear Rate (1/s) Figure 2. Rheology of simulant gel. c. Refinement criteria Since adaptive mesh refinement (AMR) uses multiple-levels mesh, care must be taken to ensure the accuracy of the results. In present method, several refinement criteria are used depending on the problem. For interfacial flows, four basic refinement criteria are applied to improve the accuracy and efficiency. There are gradient-based, value-based, curvature-based, and topology-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 spatially. 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 refinement method adapted from digital topologic theory[26] is extended and implemented to ensure that there are at least certain number of grids in the thickness direction. We call this method Topology-Oriented Adaptive Mesh Refinement (TOAMR). Figure 3(a) is the overall structure of the simulation result. Both thickness- and curvature- based refinements are used to ensure the accuracy. The fish-bone

5 like flow structure is caught. The size scale and thickness 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 interfacial cell near the impact point. From the impact point to the downstream of the liquid film, the cell level increases according to the decreased film thickness. Figure 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. 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 refinement during the simulation provides highly resolved images of the flow structures and atomization process, which will be discussed in detail in the following sections. (a) (b) (c) 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 interface 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 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 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 present simulation results. The Weber number for the three cases from top to bottom are 1549, 6195 and 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 propagate 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

6 under the effect of surface tension. Due to fluid viscosity, 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 amplitude surface waves on the liquid sheet. The damped amplitude is caused by the high viscosity of the simulant 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).

7 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 ligaments 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 impact wave. The thin regions of liquid sheet are torn off by the impact wave. The gaps contract into liquid ligaments 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 = is shown in Figure 7. Since the inlet velocity is higher, the droplet shedding at the beginning of the impingement process result in formation of a droplet cloud (see Figure 7(a) to (d)). The amplitude 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 velocity are longer and thinner as shown in Figure 8.

8 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) Figure 7. Development of flow pattern under We=12390 (interface colored by y coordinate).

9 Figure 8. Comparison of detail atomization process at We=6159 (left) and We=12390 (right). Strouhal number is defined as St=fD/U s =D/λ, where f is wave propagation frequency, D the jet diameter, U s the velocity of liquid sheet, and λ is the wavelength. The values of St under different We are plotted in Figure 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 geometrical parameters, i.e. jet diameter and impingement angle. found near the boundaries and near the impact point of the two jets. The former can be attributed to the velocity 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 regions 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 bubbles in Figure 10(c). The reduction in gel viscosity indicates that, with the same initial viscosity, the non- Newtonian liquid is easier to atomize than the viscous Newtonian fluid for the impinging jets configuration. (a) (b) 1.0 St We 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 respectively. The dark red regions indicate large shear rates. Figure 10(a) shows that large shear rates are Figure 10. Distribution of shear rate at the cross section under different Weber number. (a) We=1549; (b) We=6195; (c) We= 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 (c)

10 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 beads on a string describing a form of spherical droplets connected by thin ligaments. Figure 11(c) shows more complicated breakup process of liquid ligaments with higher inlet velocity. There are more twisted ligaments, long ligaments with dramatically changed diameters, and finer droplets. (a) (b) (c) Figure 11. Distribution of shear rate on the interface under different Weber number. (a) We=1549; (b) We=6195; (c) We= 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 results of near-field fluid dynamics. Good overall agreements 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 previous 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 configuration. The optimization of the design parameters, for example impingement angle, jet diameter, and offcenter distance of the two jets, can improve the atomization 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 U.S. Army Research Office under the Multi-University Research Initiative with contract No. W911NF The support and encouragement provided by Dr. Ralph Anthenien are gratefully acknowledged. Special thanks is due to Dr. Stéphane Popinet for allowing us to use his VOF and AMR algorithms. References 1. Bayvel, L. and Z. Orzechowski, Liquid atomization. Vol : CRC. 2. Lefebvre, A.H., Atomization and sprays. 1989: CRC. 3. Sutton, G.P. and O. Biblarz, Rocket propulsion elements. 2011: Wiley. 4. Bush, J.W.M. and A.E. Hasha, On the collision of laminar jets: fluid chains and fishbones. Journal of Fluid Mechanics, : p Anderson, W.E., H.M. Ryan, and R.J. Santoro, Impinging jet injector atomization. Liquid rocket engine combustion instability(a ), Washington, DC, American Institute of Aeronautics and Astronautics, Inc.(Progress in Astronautics and Aeronautics., : p Dombrowski, N. and P. Hooper, A study of the sprays formed by impinging jets in laminar and turbulent flow. J. Fluid Mech, (3): p Anderson, W.E., H.M. Ryan III, and R.J. Santoro, Impact Wave-Based Model of Impinging Jet Atomization. Atomization and Sprays, (7): p Inoue, C., T. Watanabe, and T. Himeno, Study on Atomization Process of Liquid Sheet Formed by Impinging Jets, in 44th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit. 2008, American Institute of Aeronautics and Astronautics. 9. Inoue, C., T. Watanabe, and T. Himeno, Liquid Sheet Dynamics and Primary Breakup Characteristics at Im-

11 pingement Type Injector, in 45th AI- AA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit. 2009, American Institute of Aeronautics and Astronautics. 10. Arienti, M., et al., Coupled Level-Set/Volume-of- Fluid Method for the Simulation of Liquid Atomization in Propulsion Device Injectors, in 46th AI- AA/ASME/SAE/ASEE Joint Propulsion Conference & 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 American Institute of Aeronautics and Astronautics. 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 Chen, X., D. Ma, and V. Yang, High-Fidelity Numerical Simulations of Impinging Jet Atomization, in 48th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, Atlanta, Georgia Fakhri, S.A.K., 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 Impinging Jet Sprays. in ICLASS 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, Popinet, S., Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries. Journal of Computational Physics, (2): p Popinet, S., An accurate adaptive solver for surfacetension-driven interfacial flows. Journal of Computational Physics, (16): p Chorin, A., On the convergence of discrete approximations to the Navier-Stokes equations. Mathematics of Computation, (106): p Brackbill, J., D. Kothe, and C. Zemach, A continuum method for modeling surface tension* 1. Journal of Computational Physics, (2): p Popinet, S. and S. Zaleski, A front-tracking algorithm for accurate representation of surface tension. International Journal for Numerical Methods in Fluids, (6): p Rahimi, S., et al., Preparation and characterization of gel propellants and simulants. AIAA paper, 2001( ). 23. Natan, B. and S. Rahimi, The status of gel propellants in year Combustion of energetic materials, 2001: p Metzner, A. and J. Reed, Flow of non newtonian fluids correlation of the laminar, transition, and turbulent flow regions. AIChE Journal, (4): p Dodge, D. and A. Metzner, Turbulent flow of nonnewtonian systems. AIChE Journal, (2): p Tchon, K.F., et al., Three-dimensional anisotropic geometric metrics based on local domain curvature and thickness. Computer-Aided Design, (2): p Taylor, G., Formation of thin flat sheets of water. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, (1296): p