10th International Symposium on Cavitation - CAV2018
Baltimore, Maryland, USA, May 14 – 16, 2018
CAV18-05060
Direct Numerical Simulations of Gas-Liquid Flows
1
Gretar Tryggvason*; 1Jiacai Lu; 2Ming Ma
1
Johns Hopkins University, Baltimore, MD, USA; 2University of Notre Dame, Notre Dame, IN, USA
Abstract
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].
*Corresponding Author, Gretar Tryggvason: gtryggv1@jhu.edu
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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.
10th International Symposium on Cavitation - CAV2018
Baltimore, Maryland, USA, May 14 – 16, 2018
CAV18-05060
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
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Figure 2. The interface between gas and liquid at
one time from a simulation of two-fluid flow
undergoing coalescence and breakup.
Figure 1. Details of the flow (bubbles and vorticity)
at one time from a simulation of hundreds of
bubbles in a turbulent channel flow.
10th International Symposium on Cavitation - CAV2018
Baltimore, Maryland, USA, May 14 – 16, 2018
CAV18-05060
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
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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.
10th International Symposium on Cavitation - CAV2018
Baltimore, Maryland, USA, May 14 – 16, 2018
CAV18-05060
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.
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].
References
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The results shown in figures 1 and 2, as well of data from many other simulations, some of which has been made
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relationships can be improved considerably by a judicious selection of nondimensional and appropriately variables,
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10th International Symposium on Cavitation - CAV2018
Baltimore, Maryland, USA, May 14 – 16, 2018
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