Dynamic Droplet Model for Two-Component Flows

There are many two-fluid mixtures of practical interest. For example, mixtures of air and water or oil and water occur frequently in environmental and industrial processes. The existing modeling capability in FLOW-3D for mixtures of two fluids is the DRIFT routine that computes the relative motion of the two components arising from buoyancy and viscous forces. This model works quite well for many situations because of its simplicity and robustness.

There are two types of two-component flows. One is a mixture of two fluids having different densities, but no sharp two-fluid interface. The other type is a single fluid, which may have a free surface, but with a variable density arising from the mixture of two incompressible components.

The dispersed component is assumed to consist of spherical droplets or bubbles. This model, however useful, requires the user to specify the average size of the dispersed material elements so that forces driving any relative motion can be computed. Specifying the size has its limitations because it is often not something that is known. Furthermore, sizes can vary in both space and time, for instance, in a mixing vessel containing an impeller it is likely that sizes are smaller near the impeller and may be further decreasing with continued mixing.

To rectify this limitation, a new model has been implemented for dynamically computing dispersed fluid droplet sizes within the DRIFT routine. The model is based on simple mechanisms for the breakup and coalescence of dispersed material controlled by the critical Weber and Capillary numbers. The new model produces sizes that vary in both space and time as they adjust to the details of the local flow.

Layout of experimental chute
Figure 1. Layout of experimental chute

A good application of the new model is air entrapment in water flowing down a spillway or sloped chute. The published thesis of K. Krammer (Development of Aerated Chute Flow, ETH, Zürich, 2004) offers an excellent example of this type of flow that includes measured data on the distribution of bubble sizes within the flow. The chute, shown in Fig. 1, was 14m long, 0.5m wide and tilted with respect to the horizontal by 5.71°. An inlet flow depth of 0.05m was used with a premixed air concentration of 17.76% and containing 10% turbulence. This flow has an inlet flow rate of 0.1755 m3/s, which corresponds to the Froude number of 10.03.

Simulations were done with the FLOW-3D advanced air entrainment model that includes buoyancy and bulking of the water-air mixture. In addition to the entrained air introduced with the inlet flow, there is also air entrained at the surface of the flow that adds volume (bulking) to the mixture, as illustrated in Fig. 2, which shows the flow in the symmetry plane along the first meter of the chute. Beyond the first meter from the inlet the flow is nearly uniform in depth and in the distribution of air and bubble sizes.

Air volume bulking
Figure 2. Center line plot of air volume bulking up fluid in first meter of the chute

In the experiment the nominal free surface of the flow was assumed to be at the location where the concentration of the air is 90% (or 10% water). In the simulation this condition is imposed by using the air escape mechanism and limiting the minimum water volume fraction to be 0.1.

The computed volume fraction of air in the cross section of the chute, at the end of the chute where the flow is nearly steady and uniform is shown in Fig. 3. A small disturbance exists at the side walls is associated with observed roll waves emanating from frictional forces at the sides of the chute.

Air concentration at end of chute
Figure 3. Air concentration at end of chute

The computed depth of flow is 0.061m, which is close to the reported depth of 0.062m. The reported average air content at the end of the chute was about 16.7%, whereas the simulation has a value of about 20%.

Experimental vs computational results
Figure 4. Computed bubble diameters at the end of the chute versus normalized depth in comparison with experimental data. Errors bars on the computational results indicate variations with time since the solution is only quasi-steady.

A comparison of simulation results with experimental data for bubble diameters versus depth, at the midpoint of the chute is shown in Fig. 4. The agreement is very good overall. Although the simulated sizes are somewhat smaller than the data near the top of the flow, the experimenters reported that sizes near the free surface were difficult to accurately measure. In any case, it is clear that an assumption of constant bubble size is not realistic and that the new dynamic droplet model has added more realism to FLOW-3D simulations of two-component flows.