Simulation of Free Surface Flows with Surface Tension with ANSYS CFX C. Kurt Svihla, Hong Xu ANSYS, Inc. Abstract Three different test cases involving free surface flows with surface tension were investigated with ANSYS CFX. Transient analyses were performed in all cases. The three cases were: (1) oscillations of a twodimensional cylindrical rod; (2) rise of liquid in a capillary; and (3) rise of an air bubble train through stagnant water. The mesh resolution, timestep, and solver settings required to obtain accurate results for the individual cases were investigated and are reported for the individual cases. Good agreement between the simulation results and results reported in the literature were obtained for the cases studied. Introduction Typically multifluid free surface flows exhibit a distinct, sharp interface between the phases. The position of this interface is normally of great interest and a number of methods have been developed to resolve and track this interface in computationa fluid dynamics (CFD) simulations of free surface problems. Annaland et al. (2005) recently reviewed a number of different methods for simulating multifluid flows with sharp interfaces listing advantages and disadvantages of each. The various methods they discuss include level set, shock-capturing, marker particle, simple line interface calculation volume-of-fluid (SLIC VOF), piecewise line interface calculation volume-of-fluid (PLIC VOF), Lattice Boltzmann, and front-tracking approaches. The goal of the different approaches is to track the position of the interface accurately without introducing excessive computational smearing. When the interface between the fluids is highly curved, then the effect of the surface tension force in the multifluid simulation becomes particularly important. Surface tension acts at the interface between two fluids. Computationally, this is awkward to deal with and the role of surface tension is generally incorporated as a continuum surface force using the approach of Brackbill et al. (1992). ANSYS CFX uses a compressive discretization scheme in both time and space to minimize the smearing of the free surface at the interface. Typically, this reduces smearing at the interface to two or three layers of cells. This is an interface capturing approach rather than an interface reconstruction approach of the type reviewed by Annaland et al.. Three different test cases for modeling transient free surface flows with surface tension have been considered. These include the oscillation of a planar rod initially deformed from a circular cross-section, rise of fluid in a capillary, and rise of a spherical air bubble train through otherwise stagnant liquid water. Procedure Oscillation of a Planar 2D Rod For this simulation, a planar 2D geometry was constructed. The initial condition for the simulation had the liquid confined to a square region. The transient simulation then modeled the oscillation of the droplet as it approached a circular cross-section.
The computational domain was 4 cm x 4 cm (planar 2D) with a thickness of 0.1 cm in the symmetry direction. The homogeneous free surface model in ANSYS CFX was used to simulate the oscillation of the rod or droplet. Buoyancy was neglected in the analysis. The interface compression level was set to 2. According to Ham and Young, (2003), the expected angular frequency of oscillation for a 2D column in the inviscid limit should be given by the expression of Lamb (1932): ω 2 = (n)(n 2-1) (γ/ρ r 3 ) where n is the mode of oscillation. Dynamic Capillary Rise For the capillary rise simulation, the experimental apparatus used by Zhmud et al. (2000) was simulated. In their paper, Zhmud et al. reported dynamic capillary rise measurements for dodecane rising through a precision bore borosilicate glass capillary. An axisymmetric 2D mesh was constructed for the simulation. The computational domain comprised an angular extent of 5 degrees. The initial condition had a flat interface between gas and liquid with the capillary extending a short distance into the liquid. The inner surface of the capillary wall had a contact angle of 17 prescribed so that liquid would rise inside it as the transient simulation proceeded. No wall adhesion parameters were defined for any of the other walls in the geometry. The homogeneous free surface model in ANSYS CFX was used to simulate the transient rise of liquid in the capillary. Buoyancy was included in the analysis with a reference density equal to that of the gas phase. In the surface tension model, the volume fraction smoothing type was set to none which proved to be important in maintaining a reasonably well-resolved curved interface. The interface compression level was set to 2. Rise of a Bubble Train through a Stagnant Liquid This simulation considered rise of a 1 mm diameter air bubble train through otherwise stagnant water. Periodic conditions were imposed at the upper and lower bounds of the domain so that rise of a train of bubbles rather than a single isolated bubble was considered. An axisymmetric 2D mesh was constructed for the simulation. The computational domain comprised an angular extent of 5 degrees. The initial condition for the simulation placed a bubble with a diameter of 1mm near the lower bound of the domain. A chart given by Grace and Weber (1982) shows that a 1 mm air bubble should remain spherical as it rises so that an axisymmetric treatment is appropriate. Bubbles much larger than this would follow helical paths as they rise which an axisymmetric treatment would not fully capture. The same chart by Grace and Weber (1982) shows that the expected terminal velocity of a 1mm diameter air bubble rising through pure water is approximately 18.1 cm/s. The homogeneous free surface model in ANSYS CFX was used to simulate the transient rise of the bubble through the liquid. Buoyancy was included in the analysis with a reference density equal to that of the liquid phase. In the surface tension model, the volume fraction smoothing type was set to none and the curvature underrelaxation factor was set to 0.5. The interface compression level was set to 2.
Analysis Oscillation of a Planar 2D Rod The simulation was run in double precision with a timestep of 0.001 s and a maximum of 30 coefficient loops per time step. The grid consisted entirely of prisms (extruded triangles). The grid was one layer thick in the symmetry direction with a thickness of 0.04 cm in that direction. A cross-sectional view is shown in Figure 1. The grid consisted of 25,262 wedge or prism elements. Figure 1. Cross-Section of the Computational Grid for the Oscillating Rod Simulation The fluid properties used were characteristic of an oil-air two phase system with a liquid density of 790 kg/m3 and a gas density of 1.2 kg/m3. The surface tension for this system was set to 0.0236 N/m. The dynamic viscosity of the gas and liquid phases were both set to 0.001 Pa s. Flow was assumed to be laminar. The initial condition for the transient simulation consisted of zero velocity and pressure with a 2 cm x 2 cm square initialized to contain liquid.
Based on the initial volume of the liquid the final radius of the circular rod should be 1.128 cm and the equilibrium pressure inside the cylindrical rod should be γ/r or 2.09 Pa. The mode of oscillation for an initial square is four so that the expected period of oscillation for the system under the imposed conditions works out to 0.178 s. Dynamic Capillary Rise For the dodecane-air system simulated for this analysis, the interfacial tension was 0.025 N/m and the contact angle was 17. The liquid density was 750 kg/m3, the liquid viscosity was 0.0017 Pa s, and the radius of the capillary was 0.10 mm. The axisymmetric mesh was constructed for simulation had an angular extent of 5 degrees and inner and outer radii of 0 mm and 5.1 mm, respectively. The length of the computational domain in the axial direction was 25 mm. The mesh consisted of 57089 hexahedral elements with 697 wedges or prisms located along the axis of the geometry. The capillary was resolved by 10 uniformly spaced cells in the radial direction. Laminar flow in both phases was assumed for the simulation. The initial condition for the simulation consisted of stationary gas and liquid phases with the interface located so as to give an initial liquid depth of 20 mm. The initial pressure was defined so as to include the hydrostatic pressure in the liquid phase. The capillary extended a short distance into the liquid phase as shown in Figure 2. Figure 2. Initial Condition for the Capillary Rise Simulation The simulation was run in double precision with a timestep of 2e-5 s and a maximum of 20 coefficient loops per timestep.
Rise of a Bubble Train through a Stagnant Liquid For the air-water system simulated for this analysis, the interfacial tension was 0.072 N/m. The liquid density was 997 kg/m 3 and the liquid viscosity was 0.000889 Pa s, and the radius of the capillary was 0.10 mm. The air density was set to 1.185 kg/m 3 and the air viscosity to 0.00001831 Pa s. For a bubble with a diameter of 1 mm and with a surface tension of 0.072 N/m, the expected pressure inside the bubble is 2γ/R or 288 Pa The axisymmetric mesh constructed for the simulation had an angular extent of 5 degrees with radial and axial dimensions of 5 mm and 20 mm, respectively. The radius of the 1 mm diameter bubble was resolved by approximately 10 cells. Overall, there were 55 cells in the radial direction and 400 cells in the axial direction for a total of 22000 elements in the computational mesh (21600 hexahedral elements and 400 wedge elements along the axis). A partial view of the computational mesh is shown in Figure 3 which shows the initial condition for volume fraction for the simulation at one of the symmetry planes for the whole radius and the lower half of the domain. This mesh was the same used by Xu and Guetari (2004a) for a similar analysis in CFX-4. Figure 3. Initial Condition for the Bubble Rise Simulation Flow was assumed to be laminar in both phases. A variable time step was used ranging from 1e-6 s at the start of the simulation to 4e-5 s at later stages. A maximum of 30 coefficient loops per timestep were used with a conservation target of 0.01 and a residual criterion of 1e-6. The initial condition for the simulation consisted of stationary gas and liquid phases with a uniform zero initial relative pressure with volume fractions set so as to define a 1 mm bubble near the bottom of the domain.
Analysis Results & Discussion Oscillation of a Planar 2D Rod Figures 4 through 8 show the oscillation in the shape of the liquid at various times during the simulation. Figure 4. Initial Shape of the Liquid Droplet Figure 5. Shape of the Liquid Droplet at 0.01 s
Figure 6. Shape of the Liquid Droplet at 0.05 s Figure 7. Shape of the Liquid Droplet at 0.09 s
Figure 8. Shape of the Liquid Droplet at 0.13 s Figure 9 shows the oscillation in the maximum height of the isosurface over the computation period of 1.1 s. The computed period of oscillation is 0.18 s which matches the expected period of 0.178 s very closely. There is about a 20% reduction in the amplitude of the oscillation from one period to the next. The droplet radius in the simulation is approaching the expected final value of 1.128 cm. Figure 9. Oscillation of the Maximum Height of the Droplet
The average pressure inside the droplet has attained a stable value of approximately 2.18 Pa after 1.1 s of simulation time which compares favorably with the expected equilibrium value of 2.09 Pa. Figure 10 shows the oscillation of the average pressure within the rod over time. The period for the pressure oscillation appears to be approximately one-half of that for the oscillation in droplet shape. Figures 11 through 15 show pressure fluctuations within the droplet at different stages of the simulation. Figure 10. Oscillation of the Average Pressure within the Droplet Figure 11. Pressure Contours at 0.49 s for the Oscillating Rod
Figure 12. Pressure Contours at 0.51 s for the Oscillating Rod Figure 13. Pressure Contours at 0.67 s for the Oscillating Rod
Figure 14. Pressure Contours at 0.75 s for the Oscillating Rod Figure 15. Pressure Contours at 0.79 s for the Oscillating Rod The droplet did remain centered within the computational domain and symmetry in the z-direction was maintained throughout the simulation.
Dynamic Capillary Rise Figures 16 and 17 show the capillary rise after 0.06 s of simulation time (Figure 17 shows the computational grid in order to display the mesh resolution). The contact angle enforced on the inner wall of the capillary is causing curvature of the interface and a meniscus to develop. It is worth noting that the liquid interface away from the immediate vicinity of the capillary tube does remain flat throughout the simulation as is displayed in Figures 16 and 17. Figure 16. Capillary Rise at a Simulation Time of 0.06 s Figure 17. Capillary Rise at a Simulation Time of 0.06 s (with superimposed grid)
Figure 18 shows a comparison of the transient capillary rise predicted by this simulation with the experimental data reported by Zhmud et al. (2000) and earlier results for this same system obtained by Xu and Guetari (2004b) with the CFX-4 code. The agreement between experiment and simulation is fairly good with the present results obtained with ANSYS CFX 10 matching experimental trends for longer simulation times than the earlier results obtained with CFX-4. Figure 18. Comparison of Experimental and Simulated Transient Capillary Rise Rise of a Bubble Train through a Stagnant Liquid As the simulation proceeds, the bubble initially remains spherical with a relatively crisp interface around the bubble perimeter. Although a zero initial pressure was set for the simulation, the pressure inside the bubble develops according to the expected value of 288 Pa. Figure 19 shows the position of the bubble after approximately 0.001 s of simulation time. The pressure inside the bubble is already near the expected value. The interface is relative crisp around the bubble perimeter. At this stage, the bubble position has been deflected only slightly from its initial position.
Figure 19. Bubble Pressure and Position after 0.001 s of Simulation Time As the bubble begins to rise, there is some smearing of the interface at the tail of the bubble although the interface remains crisp at the bubble s leading edge. If an isosurface for an air volume fraction of 0.5 is used to define the extent of the bubble, then the bubble remains approximately spherical as it begins to rise. Figures 20 and 21 show plots of volume fraction and pressure on one of the symmetry planes after 0.01 s of rise time. At this stage, the bubble has moved upward approximately 1 bubble diameter. The interface at the bubble tail has been smeared over three or four layers of cells. The pressure inside the bubble has been reduced slightly but is still close to the expected value.
Figure 20. Bubble Volume Fraction and Position after 0.01 s of Simulation Time Figure 21. Bubble Pressure and Position after 0.01 s of Simulation Time
Figure 22 shows a plot of volume fraction on one of the symmetry planes after 0.02 s of rise time. At this stage, the bubble has moved upward approximately 3.5 bubble diameters. There has been some smearing and a slight loss of material in the bubble wake. Calculations of the volume enclosed by an isosurface defined by an air volume fraction of 0.5 indicate that about 8% of the original bubble volume has been lost which corresponds to a diameter reduction of about 3%. This calculation is somewhat imprecise since the grid resolution for the bubble is fairly coarse and the circular bounds of the isovolume are approximated by a Cartesian grid. Nevertheless, both the crispness of the interface and preservation of the initial bubble size could be improved by increasing the resolution of the grid. Dijkhuizen et al. (2005) note that it has proven to be particularly difficult to simulate the motion of small bubbles with diameters on the order of 1 mm with VOF or front-tracking approaches due to parasitic currents and unacceptable loss of material due to the surface tension treatment. Viewed in this light, the current results seems encouraging especially since Dijkhuizen et al. state that their front-tracking simulations were the first published results for 1 mm bubbles with realistic fluid properties. Figure 22. Bubble Volume Fraction and Position after 0.02 s of Simulation Time Figure 23 shows a plot of the bubble s upward velocity with time. After 0.02 s, the rise velocity is in reasonably good agreement with the expected terminal rise velocity and is trending towards a stable value. The spherical bubble shape has been maintained during this simulation period. Care should be taken to examine long-term results to ensure that the bubble size remains within acceptable limits as it rises.
Figure 23. Transient Bubble Rise Velocity Conclusion In this paper, three cases of simulating free surface flows with surface tension with ANSYS CFX have been presented. Whenever possible, the results from ANSYS CFX were verified by comparison with experimental data, correlations, or theory. For the three cases considered, the results from ANSYS CFX are generally in good agreement with the expected values and behavior. References Annaland, M. van Sint, W. Dijkhuizen, N.G. Deen, and J.A.M Kuipers, 2006, Numerical Simulation of Gas Bubbles using a 3-D Front-Tracking Method, AIChe J., 99-110. Brackbill, J.U., D.B. Kothe, and C. Zemach, 1992, A Continuum Method for Modeling Surface Tension, Journal of Computational Physics, 110, 335 353. Dijkhuizen, W., E.I.V. van den Hengel, N.G. Deen, M. van Sint Annaland and J.A.M. Kuipers, 2005, Numerical Investigation of Closures for Interface Forces Acting on Single Air-Bubbles in Water Using Volume of Fluid and Front Tracking Models, Chem. Eng. Sci., 60, 6169-6175, Ham, F., and Y.N. Young, 2003, A Cartesian Adaptive Level Set Method for Two-Phase Flows, Center for Turbulence Research Annual Research Briefs, 227-237. Lamb, H., 1932, Hydrodynamics, Cambridge University Press. Xu, H. and C. Guetari, 2004a, Numerical Simulations of a Train of Air Bubbles Rising Through Stagnant Water, 2004 International ANSYS Conference, Pittsburgh.
Xu, H. and C. Guetari, 2004b, The Use of CFD to Simulate Capillary Rise and Comparison to Experimental Data, 2004 International ANSYS Conference, Pittsburgh. Zhmud, B.V., F. Tiberg, and K. Hallstensson, 2000, Dynamics of Capillary Rise, J. Colloid and Interface Science, 228, 263-269.