Direct Numerical Simulations of Gas-Liquid Flows

*Corresponding Author, Gretar Tryggvason: [email protected]

Direct Numerical Simulations of Gas-Liquid Flows 1Gretar Tryggvason*; 1Jiacai Lu; 2Ming Ma

1Johns Hopkins University, Baltimore, MD, USA; 2University of Notre Dame, Notre Dame, IN, USA

Abstract Recent progress in direct numerical simulations (DNS) of gas-liquid flows is discussed. We start by reviewing briefly DNS of cavitating and non-cavitating flows and then address two advanced topics: How to use results for bubbly flows to help generate improved models for the large-scale or average flow, and simulations of flows undergoing massive topology changes. For the first topic we have started to experiment with the use of machine learning methods to extract complex correlations from the DNS data and for the second we are exploring how to diagnose the flow and describe its structure.

Keywords: Direct numerical simulations; turbulence; machine learning; topology changes

Introduction

Direct numerical simulations (DNS) of multiphase flows, where every continuum length and time scale is resolved for a flow spanning a large range of spatial and temporal scales, have advanced significantly over the last decade and it is now possible to simulate hundreds of bubbles in turbulent flows for a sufficiently long time so that meaningful statistical description of the flow can be obtained. Although a few DNS studies have been done for cavitating flows, most progress has been made in the context of incompressible fluids. DNS have provided significant new insight into the dynamics of such flows and the influence of bubble deformability, void fraction and surfactants, on the overall flow structure. Cavitation can be simulated in two ways. The most complete approach is to treat it as constant temperature boiling. Significant progress has been made in simulations of boiling, although nucleate boiling still poses some challenges (see [1], for recent progress). The other approach is to simply specify the pressure inside the bubble when solving the Navier-Stokes equations. The pressure in the liquid is solved to enforce incompressibility but the pressure inside the bubbles is set to either the vapor pressure or a pressure based on the bubble volume for gas bubbles. This approach has been used to simulate the shock propagation through bubbly flows for both two- and three-dimensional flows and assuming both vapor and gas bubbles [2,3], and by [4] who compared the results for three-dimensional flow for gas bubbles with predictions by models for the average flow based on the Rayleigh-Plesset equation. The results were also used to develop an improved model. Overall the results suggest that this simplified model (assuming a constant pressure in the bubbles) works relatively well and captures the change of size and shape of bubbles as the pressure changes. A simulation of slightly more complex flow, but for two-dimensions only, can be found in [5]. This approach is a relatively straightforward extension of simulations of flows with incompressible bubbles and it should, in particular, be possible to simulate cavitation bubbles in turbulent flows. For other examples of simulations of the collapse of cavitation bubbles in inviscid compressible flows see [6], which describes what was, at the time, the largest fluid dynamics simulation ever done in terms of number of grid points used to resolve the flow field. Studies of incompressible bubbles in turbulent flows include a demonstration of the importance of deformability for bubble induced drag reduction in turbulent flows [6], and several investigations of turbulent bubbly flows in vertical channels (see [8,9], for example). Although most of those simulations are still at modest Reynolds numbers, a few investigators have started to explore how the data can be used to help improve two-fluid models for the average flow [10,11]. Others have initiated work on using DNS data to help develop large eddy simulation (LES) models for multiphase flows [12-14].


The most significant outcome of DNS studies is, of course, the enormous amount of data that such simulations produce and the question of how to use this data to assist with the generation of reduced order models of one sort or another is emerging as a major challenge and we discuss briefly how machine learning can be used to extract closure terms for reduced order or averaged models of two-phase flows. The second challenge is that many gas-liquid flows are very complex and the phase boundary forms a dynamic structure that cannot be described as bubbles and drops. We have recently started to examine such flows and how they can be characterized.

Numerical Approaches

Most simulations of multiphase flows rely on the so called one-fluid or one-field formulation of the governing equations, where one set of equations is solved for the whole flow field, using a stationary grid. The different fluids or phases are identified by an index function and the various material properties are set based of the index function. Forces and other effects concentrated on the interface are included using delta functions smoothed onto the fixed grid. This allows the use of flow solvers similar to those used for single-phase flow, although they must accommodate sharp changes in the material properties and this can pose additional challenges. Nevertheless, most approaches to simulating multiphase flows use similar solvers for the flow field and the difference between the various methods is how the index function is advected and surface tension is included. Popular methods to advect the index function include the volume-of-fluid and the level set methods, and their many extensions, where the index function is advected directly. In our studies we use a front tracking method where the interface between the different fluids/phases is marked by connected marker points that are advected by the flow. The index function is constructed from the location of the marker points and the marker points are also used to find the surface force. This hybrid approach, using Lagrangian markers to advect the interface and a stationary Eularian grid to solve the governing equations, has been found to be both versatile and accurate. For more details, as well as discussion of the various methods for DNS of multiphase flows, see [15, 16].

Results

Figure 1 shows a part of the flow field at one time from a simulation of a few hundred clean bubbles rising in a vertical turbulent channel flow. The bubbles and the vorticity, visualized using the lambda-2 method are shown and the color indicates the direction of vorticity, with red and blue indicating the streamwise direction but rotations in the opposite directions. The results, described in more detail in [17] show that small bubbles migrate toward the walls but the larger deformable ones stay in the center of the channel. In this simulation the bubbles are not allowed

Figure 1. Details of the flow (bubbles and vorticity) at one time from a simulation of hundreds of

bubbles in a turbulent channel flow.

Figure 2. The interface between gas and liquid at one time from a simulation of two-fluid flow

undergoing coalescence and breakup.


to coalesce or break apart. In figure 2 we show one frame from a simulation of flows undergoing massive topology changes. Here, several deformable bubbles (high Weber number) are placed in a turbulent channel flow at a sufficiently high void fraction so that the bubbles collide and the liquid film between them becomes very thin. This film is ruptured at a predetermined thickness and the bubbles are allowed to coalesce. For low Weber numbers the bubbles continue to coalesce, eventually forming one large bubble. At high Weber numbers, on the other hand, the large bubbles break up again, sometimes undergoing repeated coalescence and breakup. For the parameters used for the simulation in figure 2, the statistically steady state consists mostly of bubbles of roughly two sizes. The evolution of the various integral quantities, such as the average flow rate, wall-shear, and interface area are monitored and compared for different governing parameters, and the microstructure at statistically steady state is quantified using low order probability functions. This simulation is described in more detail in [18] where the various ways to characterize the flow are explored. Describing the complex interface topology in statistical ways

Figure 3. The correlation between the horizontal gas fluxes and various variables describing the average state of the flow is shown in the top frames. On the left the data contains 18 variables and on the right it contains 6

variables. In the bottom frame the Gini index shows what variables are most important.


that convey the average structure and range of scales, poses considerable challenges and it is not clear at the present time what the optimal description is. The results shown in figures 1 and 2, as well of data from many other simulations, some of which has been made available to other researchers [11], allows us to collect essentially any average and statistical quantity that we desire. This allows us to explore how closure terms, for example, depend on the various average quantities. As the flow becomes complex, finding the best correlations becomes complex and we have been exploring how the data can be analyzed in advanced ways. We have, in particular, used neural networks to find correlations between closure terms in simple models of the average flow and resolved quantities for simple laminar flows with nearly spherical bubbles [19,20]. Most recently, we have also started to use statistical learning to help determine what quantities best describe the unresolved fields in turbulent bubbly flows. Those include both variables commonly used for the modeling of turbulent flows, such as the kinetic energy and dissipation rate, as well as quantities describing the interfacial microstructure such as area concentration and various area projections. The goal is to find the variables that best correlate the closure terms and which would therefore be natural candidates for modeling. Figure 3 shows an example of such an analysis for nearly spherical bubbles in turbulent upflow in a vertical channel. The top frames show how well the horizontal gas flux correlates with the velocity gradient, the void fraction, the invariant of the velocity gradient, and various other quantities. To investigate which of these variables is most important we plot the Gini index in the frame on the bottom, which shows that the flux depends most strongly on the velocity gradient and the velocity invariance and then the dependency drops of relatively slowly. The frame in the top right corner shows the correlation of the gas fluxes with the top six variables from the frame on the bottom and while it is not as good as in the left frame, the correlation is still reasonably strong. We have examined a few other cases in the same way, but often find that the best variables depend on the particular flow regime that we are examining. It is likely that the relationships can be improved considerably by a judicious selection of nondimensional and appropriately variables, but we have not done so yet. We do, however, believe that the use of machine learning to help select the appropriate variables for modeling and closure holds considerable promise.

Conclusion

Direct numerical simulations of multiphase flows, where verified solutions of validated equations are solved for flows spanning a range of temporal and spatial scales, are now capable of providing results that can, at least in principle, be used to help generate improved models for simulations of the average flow and large eddy motion of bubbly turbulent flows. So far, relatively little has been done for cavitating flows, but those studies done so far suggest that most of the methodology carries over with only modest changes. As simulations of bubbly turbulent flows become almost routine, new challenges are emerging and we have discussed two of those. The first is the need for more sophisticated methods to analyze the data generated by the simulations and the second is how to extend the simulations to more complex situations, such as multiphase turbulent flows where the phase boundary has a complicated structure and the topology changes due to mergers and breakups. Machine learning is emerging as one possible way to help with the former and various multiscale strategies are likely to be required for the latter. The success of DNS of bubbly flows in the past has also opened up the possibilities of exploring more complex physics, such as heat transfer and surfactants in turbulent bubbly flows [21,22].

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Direct Numerical Simulations of Gas-Liquid Flows