# Two-phase Flow

Direct Numerical Simulations of two-phase flows can be performed in TrioCFD using a Front Tracking algorithm mixed with VoF capabilities. The algorithm was developped in the PhD work of Mathieu (2003) (see Publication section) and further extended later on to implement the Ghost Fluid Method to deal with velocity and temperature gradient discontinuities in the case of phase-change. A lot of application were performed during several PhDs in the group. The gallery below illustrates some of the most recent applications.

**Front-Tracking (Discontinuous). **

*When to use it?*

TrioCFD has a Front-Tracking (FT) module to perform Direct Numerical Simulations of two-phase flows with explicit interface tracking. It is applicable in 2D, 2D axisymmetric or 3D geometries, using either VDF or VEF discretization. It can be used to fully-resolve incompressible two-phase flows with constant properties in each phase. Contact line can be considered with a prescribed contact angle (possibly dependent on the position). Coalescence and break-up are correctly handled with adaptation of the mesh topology to avoid inter-penetrations. Phase-change can be computed based on the temperature gradient in each phase, considering a constant interfacial temperature. Chemistry can also be considered in the phases with constituent transport and an instantaneous reaction rate.

Regarding turbulent fluctuations, the FT module can be used in conjunction with LES models (WALE or Smagorinsky) and wall law can be used in the liquid.

The main limitation in geometry is that interfaces cannot properly cross through periodic boundary conditions yet.

*Model's description and specificities :*

The baseline of the FT method is similar to the pioneering work of Tryggvason et al. with the notable difference that the physical properties (density, viscosity, heat mass capacity and conductivity) are kept sharp through the interface crossing (no smoothing tanh function with a thickness larger than the mesh size). Spurious currents can be limited by the use of an original formulation (Mathieu, 2003).

Surface tension and phase-change are considered with simple surface rheology (no surfactants, constant saturation temperature). Phase-change is then fully determined by the temperature gradient in each phase; the jump is considered accurately by a Ghost-Fluid Method (GFM). Apart from that, the one-fluid formalism is used to resolve mass and momentum on an Eulerian grid. The interface is described by an explicit mesh of connected markers that are transported by a n-linear interpolation of the Eulerian velocity field. Mass conservation is recovered by coupling the FT transport with the transport of a colour-function on the Eulerian grid. A distance function is also built from the interface position to be used into the GFM. Change in mesh topology (coalescence, break-up) are based on Juric algorithms.

*Examples :*

Examples of applications can be found in the gallery for two-phase flows.

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**Front-Tracking(IJK). **

*When to use it ?*

The IJK module in TrioCFD relies on the Front_tracking_discontinu module for all the FT mesh management. IJK (reference to the three components of a Cartesian coordinate system) is optimized for direct numerical simulations in parallelepipedic domains, discretized on a structured Cartesian grid. Its simplified vectorization allows significant performance gains, in particular thanks to the multigrid solver for the resolution of the pressure. This module is used to perform channel flow calculations (wall boundary conditions available in the z direction), as well as periodic calculations. The periodicity for the Front-Tracking interfaces is ensured with specific algorithm, and allows to study stationary states. The IJK Front-Tracking module is mature for adiabatic calculations of bubbly flows, and the resolution of energy conservation is in development. Front-Tracking IJK is specially designed for bubbly flow studies (neither free surfaces, nor droplets) at low void fraction (<10%). To date, there is no model for fragmentation, coalescence or wall boiling.

The conditions to use Front-Tracking IJK:

- Small parallelepipedic domain.

- At least two periodicity directions (x,y): ideal for stationary states studies.

- Ideal for adiabatic flows at low void fraction.

*Model's description and specificities :*

The Front-Tracking IJK module of TrioCFD resolves the one-fluid equations of Kataoka (1986) as written for channel up-flow by Lu & Tryggvason (2008). Following the proposal of Tryggvason et al. (2003), a front-tracking method is used to solve the set of equations in the whole computational domain, including both the gas and liquid phases. The interface is followed by connected marker points. The Lagrangian markers are advected by the velocity field interpolated from the Eulerian grid. In order to preserve the mesh quality and to limit the need for a smoothing algorithm, only the normal component of the velocity field is used in the marker transport. After transport, the front is used to update the phase indicator function, the density and the viscosity at each Eulerian grid point. The Navier–Stokes equations are then solved by a projection method (Puckett et al. 1997) using fourth order central differentiation for evaluation of the convective and diffusive terms on a fixed, staggered Cartesian grid. Fractional time stepping leads to a third-order Runge–Kutta scheme (Williamson 1980). In the two-step prediction–correction algorithm, a surface tension source is added to the main flow source term and to the evaluation of the convection and diffusion operators in order to obtain the predicted velocity (see Mathieu (2003) for further information). Then, an elliptic pressure equation is solved by an algebraic multigrid method to impose a divergence-free velocity field.

*Examples of work carried out using the Front-Tracking IJK module of TrioCFD :*

DU CLUZEAU A., BOIS G., LEONI N., & TOUTANT A. (2022). Analysis and modeling of bubble-induced agitation from direct numerical simulation of homogeneous bubbly flows. *Physical Review Fluids*, *7*(4), 044604.

DU CLUZEAU A., BOIS G., TOUTANT A., & MARTINEZ, J. M. (2020). On bubble forces in turbulent channel flows from direct numerical simulations. *Journal of Fluid Mechanics*, *882*.

DU CLUZEAU A., BOIS G., & TOUTANT A. (2019). Analysis and modelling of Reynolds stresses in turbulent bubbly up-flows from direct numerical simulations. *Journal of Fluid Mechanics*, *866*, 132-168.

DUPUY D., TOUTANT A. & BATAILLE F. (2018) Turbulence kinetic energy exchanges in flows with highly variable fluid properties. J. Fluid Mech. 834, 5–54.

BOIS G., DU CLUZEAU A., TOUTANT A. & MARTINEZ J.-M. (2017) DNS of turbulent bubbly flows in plane channels using the front-tracking algorithm of TrioCFD. In Proceedings of the ASME Fluid Engineering Division Summer Meeting & Multiphase Flow Technical Committee, ASME.

BOIS G., FAUCHET G. & TOUTANT A. (2016) DNS of a turbulent steam/water bubbly flow in a vertical channel. In Proceedings of the 9th International Conference on Multiphase Flows (ICMF2016), ICMF.

TOUTANT A., LABOURASSE E., LEBAIGUE O. & SIMONIN O. (2008) DNS of the interaction between a deformable buoyant bubble and a spatially decaying turbulence: a priori tests for LES two-phase flow modelling. Comput. Fluids 37 (7), 877–886.

*References :*

JAMET D. & LEBAIGUE O. (2009) Jump conditions for filtered quantities at an under-resolved discontinuous interface. Part 1: theoretical development. Intl J. Multiphase Flow 35 (12), 1100–1118.

LU J. & TRYGGVASON G. (2008) Effect of bubble deformability in turbulent bubbly upflow in a vertical channel. Phys. Fluids 20, 040701.

TRYGGVASON G., BUNNER B., ESMAEELI A. & AL-RAWAHI N. (2003) Computations of multiphase flows. Adv. Appl. Mech. 39 (C), 81–120.

MATHIEU B. (2003) Etudes physique, expérimentale et numérique des mécanismes de base intervenant dans les écoulements diphasiques en micro-fluidique. PhD thesis.

PUCKETT E. G., ALMGREN A. S., BELL J. B., MARCUS D. L. & RIDER W. J. (1997) A high-order projection method for tracking fluid interfaces in variable density incompressible flows. J. Comput. Phys. 130, 269–282.

KATAOKA I. & SERIZAWA A. (1989) Basic equations of turbulence in gas-liquid two-phase flow. Intl J. Multiphase Flow 15 (5), 843–855.

KATAOKA I. (1986) Local instant formulation of two-phase flow. Intl J. Multiphase Flow 12 (5), 745–758.

WILLIAMSON J. H. (1980) Low-storage Runge–Kutta schemes. J. Comput. Phys. 35 (1), 48–56.

**ReportageIllustrations of some use of the Front Tracking Algorithm**

Some DNS results for the cases of turbulent bubbly flows are available in the FT-Database page.