Gimenez et al 2018b - Revision history

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-Published in Computers & Fluids, Vol. 141, pp. 90-104, 2016 (preprint)<br />
+Published in Computers & Fluids, Vol. 141, pp. 90-104, 2016 <br />
 doi: 10.1016/j.compfluid.2016.04.026

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 ==Abstract==
-In previous works <span id='citeF-1'></span><span id='citeF-2'></span>[[#cite-1|[1,2]]], the authors have presented a highly efficient extension of the Particle Finite Element Method, called PFEM-2, to solve two-phase flows. The methodology which uses X-IVS<span id='citeF-3'></span>[[#cite-3|[3]]] to treat convection terms allowing large time-steps was validated for problems where the gravity forces and/or the inertial forces dominate the flow. Although that is the target range of problems to solve with PFEM-2, most of real problems that fall in these categories also includes other flow regimes in certain regions of the domain. Maybe the most common secondary regime is when the surface tension dominates, as an example when drops or bubbles are released from the main flow, and this feature must be taken into account in any complete numerical strategy.
+In previous works <span id='citeF-1'></span><span id='citeF-2'></span>[[#cite-1|[1,2]]], the authors have presented a highly efficient extension of the Particle Finite Element Method, called PFEM-2, to solve two-phase flows. The methodology which uses X-IVS<span id='citeF-3'></span> [[#cite-3|[3]]] to treat convection terms allowing large time-steps was validated for problems where the gravity forces and/or the inertial forces dominate the flow. Although that is the target range of problems to solve with PFEM-2, most of real problems that fall in these categories also includes other flow regimes in certain regions of the domain. Maybe the most common secondary regime is when the surface tension dominates, as an example when drops or bubbles are released from the main flow, and this feature must be taken into account in any complete numerical strategy.
 Attending to that, in this work the treatment of the surface tension to PFEM-2 is included. An implicit CSF methodology is employed together with a coupling between the marker function with a Level Set function to obtain a smooth representation of the normal of the interface which allows an accurate curvature calculation. Examples for curvature calculation and isolated bubbles and drops are presented where the accuracy and the computational efficiency are analyzed and contrasted with other numerical methodologies. Finally, a simulation of a jet atomization is analyzed. This case presents the above mentioned features: it is a inertia-dominant flow with a surface tension phenomena on drops and ligaments break up that can not be neglected.

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 ==Abstract==

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 ==Acknowledgments==
-Authors gratefully thank the help and contributions of Dr. Santiago Márquez Damián and Dr. Pedro Morin. Simulations were done in the cluster ''Bora'' at the CIMEC-UNL-Argentina and in the ''Mendieta'' cluster of the Centro de Computación de Alto Desempeño CCAD-UNC-Argentina.  Also, authors want to thank to CONICET, Universidad Nacional del Litoral and ANPCyT for their financial support (grants PICT 0830 (2013), PIP-2012 GI 11220110100331 and CAI+D 2011 501 201101 00435 LI. J. Gimenez gratefully acknowledges the support of the Argentinian National Scientific and Technical Research Council (CONICET) through a type-II doctoral scholarship and the program ERASMUS MUNDUS action 2 ARCOIRIS project for financial support through a six-month doctoral scholarship. This work was also partially supported by the European Research Council under the Advanced Grant: ERC-2009-AdG  ‘‘Real Time Computational Mechanics Techniques for Multi-Fluid Problems’’.
+Authors gratefully thank the help and contributions of Dr. Santiago Márquez Damián and Dr. Pedro Morin. Simulations were done in the cluster ''Bora'' at the CIMEC-UNL-Argentina and in the ''Mendieta'' cluster of the Centro de Computación de Alto Desempeño CCAD-UNC-Argentina.  Also, authors want to thank to CONICET, Universidad Nacional del Litoral and ANPCyT for their financial support (grants PICT 0830 (2013), PIP-2012 GI 11220110100331 and CAI+D 2011 501 201101 00435 LI. J. Gimenez gratefully acknowledges the support of the Argentinian National Scientific and Technical Research Council (CONICET) through a type-II doctoral scholarship and the program ERASMUS MUNDUS action 2 ARCOIRIS project for financial support through a six-month doctoral scholarship. This work was also partially supported by the European Research Council under the Advanced Grant: ERC-2009-AdG  ‘‘Real Time Computational Mechanics Techniques for Multi-Fluid Problems’’ and by the ERC Proof of Concept FORECAST (Ref. 664910) under H2020 Programme.
 ==Bibliography==

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 {|  class="floating_tableSCP wikitable" style="text-align: center; margin: 1em auto;min-width:50%;"
-|+ style="font-size: 75%;" |<span id='table-2'></span>Table. 2 Pressure jump and parasitic currents comparison between PFEM-2 and the simulations of <span id='citeF-42'></span>[[#cite-42|[42]]].
+|+ style="font-size: 75%;" |<span id='table-2'></span>Table. 2 Pressure jump and parasitic currents comparison between PFEM-2 and the simulations of <span id='citeF-44'></span>[[#cite-44|[44]]].
 |- style="border-top: 2px solid;"
 | style="border-left: 2px solid;border-right: 2px solid;" |  Method

← Older revision	Revision as of 14:15, 10 October 2018
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 ===3.1 Rigid Body Rotation of Zalesak's Disk===
-This test consists in the advection of a region composed of a circle with a slot []Zalesak79. If the interface tracking is accurate enough, after several revolutions, the shape must remain identical. The computational domain employed is <math display="inline">\Omega \in R^2 : [0; 100]\times [0; 100]</math>. The advected region is a circle centered at <math display="inline">(50; 75)</math> with a radius of <math display="inline">15</math> and a slot of width <math display="inline">5</math> and height <math display="inline">25</math>. The velocity field is a rigid body rotation around the center of the domain with a period of 628 time units:
+This test consists in the advection of a region composed of a circle with a slot <span id='citeF-35'></span>[[#cite-35|[35]]]. If the interface tracking is accurate enough, after several revolutions, the shape must remain identical. The computational domain employed is <math display="inline">\Omega \in R^2 : [0; 100]\times [0; 100]</math>. The advected region is a circle centered at <math display="inline">(50; 75)</math> with a radius of <math display="inline">15</math> and a slot of width <math display="inline">5</math> and height <math display="inline">25</math>. The velocity field is a rigid body rotation around the center of the domain with a period of 628 time units:
 {| class="formulaSCP" style="width: 100%; text-align: left;"
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 ===3.2 Single Vortex Case===
-While Zalesak's disk test is a good indicator of numerical diffusion in an interface-capturing method, it does not test the ability to preserve small scale structures of the fluid flow. A well known test to evaluate the ability of the method to solve structures of different sizes and their evolution is given by the ''vortex-in-a-box'' problem introduced by Puckett et al. []Puckett97. The difficulty of this tests is that requires the solution of an interface stretching problem. The computational domain is <math display="inline">\Omega \in R^2 : [0; 1]\times [0; 1]</math>, where the interest region is a circle centered at <math display="inline">(0.5; 0.75)</math> with a radius of <math display="inline">0.15</math>, advected with a velocity field defined by the stream function
+While Zalesak's disk test is a good indicator of numerical diffusion in an interface-capturing method, it does not test the ability to preserve small scale structures of the fluid flow. A well known test to evaluate the ability of the method to solve structures of different sizes and their evolution is given by the ''vortex-in-a-box'' problem introduced by Puckett et al. <span id='citeF-36'></span>[[#cite-36|[36]]]. The difficulty of this tests is that requires the solution of an interface stretching problem. The computational domain is <math display="inline">\Omega \in R^2 : [0; 1]\times [0; 1]</math>, where the interest region is a circle centered at <math display="inline">(0.5; 0.75)</math> with a radius of <math display="inline">0.15</math>, advected with a velocity field defined by the stream function
 {| class="formulaSCP" style="width: 100%; text-align: left;"
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 ===3.3 LeVeque Deformation Case===
-LeVeque <span id='citeF-35'></span>[[#cite-35|[35]]] proposed a three dimensional incompressible flow field which combines a deformation in the <math display="inline">x-y</math> plane with one in the <math display="inline">x-z</math> plane. This problem can be considered an extension of the previous case, requiring the correct capturing of the stretching phenomenon in three dimensions.
+LeVeque <span id='citeF-37'></span>[[#cite-37|[37]]] proposed a three dimensional incompressible flow field which combines a deformation in the <math display="inline">x-y</math> plane with one in the <math display="inline">x-z</math> plane. This problem can be considered an extension of the previous case, requiring the correct capturing of the stretching phenomenon in three dimensions.
 * The computational domain is <math display="inline">\Omega \in R^2 : [0; 1]\times [0; 1]\times [0; 1]</math>.
@@ Line 467: / Line 467: @@
 * Courant number used in simulations is aproximately <math display="inline">CFL=4</math>.
-Since the flow is reversed for <math display="inline">t>T/2</math>, after one period the function must return to its original shape. Figure [[#img-4|4]] shows that PFEM-2 successfully recovers almost the initial shape, which is a very complicated task for other numerical strategies <span id='citeF-36'></span><span id='citeF-37'></span>[[#cite-36|[36,37]]].
+Since the flow is reversed for <math display="inline">t>T/2</math>, after one period the function must return to its original shape. Figure [[#img-4|4]] shows that PFEM-2 successfully recovers almost the initial shape, which is a very complicated task for other numerical strategies <span id='citeF-38'></span><span id='citeF-39'></span>[[#cite-38|[38,39]]].
 <div id='img-4a'></div>

@@ Line 1,098: / Line 1,098: @@
 <div id="cite-37"></div>
-[[#citeF-37|[37]]] R. LeVeque. (1996) "High-resolution conservative algorithms for advection in incompressible flow", Volume 33(2). SIAM J. Numer. Anal. 627 - 665
+[[#citeF-37|[37]]] R. LeVeque. (1996) "High-resolution conservative algorithms for advection in incompressible flow", Volume 33(2). SIAM J. Numer. Anal. 627&#8211;665
 <div id="cite-38"></div>