Computational Fluid Dynamics for nuclear applications: from CFD to multi-scale CMFD

Transcription

1 Nuclear Engineering and Design 235 (2005) Computational Fluid Dynamics for nuclear applications: from CFD to multi-scale CMFD G. Yadigaroglu Swiss Federal Institute of Technology-Zurich (ETHZ), Nuclear Engineering Laboratory, ETH-Zentrum, CLT CH-8092 Zurich, Switzerland Received 17 November 2003; received in revised form 1 June 2004; accepted 31 August 2004 Abstract New trends in computational methods for nuclear reactor thermal hydraulics are discussed; traditionally, these have been based on the two-fluid model. Although CFD computations for single phase flows are commonplace, Computational Multi-Fluid Dynamics (CMFD) is still under development. One-fluid methods coupled with interface tracking techniques provide interesting opportunities and enlarge the scope of problems that can be solved. For certain problems, one may have to conduct cascades of computations at increasingly finer scales to resolve all issues. The case study of condensation of steam/air mixtures injected from a downward-facing vent into a pool of water and a proposed CMFD initiative to numerically model Critical Heat Flux (CHF) illustrate such cascades. For the venting problem, a variety of tools are used: a system code for system behaviour; an interface-tracking method (Volume of Fluid, VOF) to examine the behaviour of large bubbles; direct-contact condensation can be treated either by Direct Numerical Simulation (DNS) or by analytical methods Elsevier B.V. All rights reserved. 1. Introduction Thermal hydraulic computations for the design and the simulation of transients in Light Water Reactors (LWR) have been conducted the last decades with mainly one-dimensional (1D), two-phase flow tools; the large system codes that have been developed and continuously improved during this time have been the major, universally used tools. Tel.: ; fax: address: yadi@ethz.ch. The very first two- or multiphase system and component analysis system codes were based on the equal-velocities, equal-phase-temperatures, homogeneous-equilibrium model; RELAP4/MOD6 was a prominent example in this category (EG&G, 1978). In reality, most two-phase flows of interest are far from homogeneous. Separated-flow models, where the two phases are allowed to have different average velocities (and temperatures) followed; in this case, the average-velocity ratio is typically derived from some empirical correlation. The drift-flux model (Zuber and Findlay, 1965), characterizing this velocity ratio with two parameters having a clear physical significance /$ see front matter 2004 Elsevier B.V. All rights reserved. doi: /j.nucengdes

2 154 G. Yadigaroglu / Nuclear Engineering and Design 235 (2005) became the dominant, almost universal tool. There are cases, however, where the velocity ratio cannot be determined from the local, necessarily mixture-based, flow conditions, the phases are no longer intimately coupled, and the approach breaks down; this is the case, for example, when the two phases flow in opposite directions, each driven by a different driving force. The two-fluid (six-equation) model was introduced to deal with these situations; it is based on the so-called interpenetrating-media approach and is widely used today. This is the level of development currently implemented and used in most system codes, such as the RELAP5 (Carlson et al., 1990; RELAP5 Code Development Team, 1995) and TRAC (LANL, 1986) families originating from the US and CATHARE (Micaelli et al., 1988) from France. The United States Nuclear Regulatory Commission is producing the USNRC consolidated code based on the previous US codes (Mahaffy et al., 2000); advanced code developments are also taking place in Europe and elsewhere. As noted above, most of the two-phase safety analysis work is still performed today with 1D system codes: all flows are assumed to take place in 1D ducts and are described with their cross-sectional-average parameters, such as void fraction, velocity, enthalpy, etc. There are situations, however, e.g., steam water flows in an open core, flows through the steam generator tube bundle, flows in the downcomer of PWRs, etc. that are clearly three-dimensional (3D). Developments are needed in this area, partly to clarify certain issues and partly to remove conservatisms imposed by the use of 1D tools. As an example, we can cite the analysis of counter-current flow limitations in the downcomer during the refill phase of PWRs where penalizing assumptions are made in the absence of 3D computations; see for example, Weiss et al. (1986) and details of the 2D/3D program by Damerell and Simons (1993). One should also note that in the two- (or multi-)fluid, interpenetrating-media system codes turbulence is usually ignored The needs, the challenges and the opportunities The nuclear thermal hydraulics community is facing today interesting challenges. These include the creation of the new computational tools that will be used for improved and more detailed analysis of the present as well as the new generations of reactors. Current trends are towards multi-dimensional, -scale, -physics approaches for such analyses. Such computational developments are possible now, thanks to the opportunities offered by the tremendous increase in computing power, the maturity of state-of-the-art Computational Fluid Dynamics (CFD) methods (still applied today mainly to single-phase flow problems) and the emergence of Computational Multi-Fluid Dynamics (CMFD) methods (Yadigaroglu, 2003a). If such developments are pursued vigorously and successfully, it is likely that we will be doing simulations in the future with increasingly sophisticated numerical methods and modelling approaches and in much greater detail than in the past. These could lead to improvements in the design of the new generations of reactors and to the elimination or reduction of certain remaining uncertainties in safety analysis (Bestion, 1992) Scope of this paper Papers dealing with the trends in numerical simulation for LWR safety were presented at the FISA (2001) conference (Yadigaroglu et al., 2003) and by Yadigaroglu and Lakehal (2003). The first paper was essentially limited to the application of CFD methods to single-phase flow situations in the primary system and the containment. We also illustrated there the use of cascades of CFD studies conducted with increasing degree of sophistication and detail to clarify key issues. In the present paper, we move from single-phase CFD to Computational Multi-Fluid Dynamics applications. We also discuss and illustrate with a case study the approach consisting of addressing complex multiphase flow problems involving a range of space and time scales with cascades of computations at different scales now, the multi-scale approach (Yadigaroglu, 2003b). Although the development of three-dimensional multiphase system simulation tools (system codes) needed to address some of the situations mentioned above is a very important issue, it is not considered here. We can only say that the introduction of CMFD methods will be an automatic step in this direction, since such tools are by their nature multi-dimensional rather than 1D. It is evident that the development of new computational tools must be accompanied by experimental

3 G. Yadigaroglu / Nuclear Engineering and Design 235 (2005) work that will provide the data needed for validation of the new tools. This is a difficult task, since detailed, often 3D data will be needed. To collect such data, the experimenters would have to take advantage of advances in instrumentation, miniaturization and coupling of the sensors with sophisticated data processing equipment and software. Although the experimental work is certainly very important, it is beyond the scope of this paper that concentrates on the computational aspects. 2. Two-fluid versus one-fluid, interfacial-tracking formulations It is worth recalling the basic premise of the twofluid model at this point. The two-fluid conservation equations are based on an averaging procedure that allows both phases to co-exist at any point, according to a certain phase-indicator function or essentially a probability that leads to the definition of the local instantaneous void fraction: this is also referred to as the interpenetrating media approach. Conservation equations are written for each phase; these contain phase interaction terms that specify the mass, momentum and heat exchanges between the phases. The velocity ratio and any thermal disequilibrium between the phases are no longer specified externally via correlations or assumptions but determined by the momentum and energy exchanges between the phases and the walls. With the two-fluid model, also referred to as the six-equation approach (or more generally the multifluid model), each phase, controlled by its own conservation equations (three for each field or phase), moves and develops independently. Although the presence of the interfaces has been considered during the local averaging process (and led to the definition of the local interfacial area concentration that provides the area for the exchanges of mass, momentum and energy between the phases), the characteristics of the interfaces (their exact shape and position) are lost with the interpenetrating-media, two-fluid formulation. The topology of the phases cannot be obtained and consequently the flow regimes cannot be determined, except by correlation with the average flow conditions; the two-fluid 1D model cannot tell if, for example, at a 30% void fraction, the flow is stratified or bubbly. The addition of an interfacial area transport equation to the two-fluid equations (see Ishii, 2004) is a partial remedy to this problem since the detailed modelling of bubble interactions provides some information of the flow regime and the distribution of the phases; developments in this area are underway. The absence of topological information about the interfaces is not a shortcoming in many two-phase flow problems, but there are situations where the two phases are sharply separated (at a large scale, such as the scale of the channel) and full understanding of the situation requires knowledge of the position and geometry of the interface. This could be, for example, the case of injection of subcooled water in a pipe with stratified flow; clearly one needs to know the characteristics of the steam water interfaces to estimate the rate of condensation taking place there. The injection of a large bubble from a vent is another situation where the shape and extent of the liquid gas interface are important; we will deal with this problem as a case study in this paper. Although the two-fluid model could, in principle, deal with the vent discharge and similar problems, in practice it cannot. Indeed, one could imagine starting the vent flow problem with the volume occupied by gas characterized as a region of void fraction one, and the liquid volume as a region of void fraction zero. Numerical diffusion will very quickly mix the two phases, however, and the interface will lose its sharpness and disappear. The implementation of interface tracking methods, discussed next, is necessary to get solutions for such problems. There are also cases where prediction of the location and topology of the phases, leading essentially to the definition of the flow pattern is needed. Other situations that are good candidates for application of interface tracking methods are those for which the stability of the interface plays an important role: the stability and break-up of jets are good examples. 3. One-fluid formulation and interface tracking methods As mentioned above, tracking of the interfaces is necessary in certain situations. The various interface tracking methods are typically associated with a onefluid description of the two-phase flow: while in the two-fluid, interpenetrating-media formulation the conservation equations were appropriately averaged (for example, over volume) for each phase and sets of con-

4 156 G. Yadigaroglu / Nuclear Engineering and Design 235 (2005) Fig. 1. The classical two-fluid vs. the one-fluid/interfacial-tracking approach. servation equations for each phase resulted, in the onefluid formulation, a unique set of conservation equations is used for the entire computational domain, but the fluid properties such as density and viscosity vary sharply when we move from one phase into the other. The differences between the two approaches are illustrated in Fig. 1. The position of the interface is tracked first using a variety of procedures (e.g., see Lakehal et al., 2002). Interface tracking methods can be Lagrangian or Eulerian. The most frequently employed Eulerian interface tracking methods are the Volume of Fluid (VOF) method (e.g., Hirt and Nichols, 1981; Rider and Kothe, 1998) and the Level Set (LS) method (e.g., Osher and Sethian, 1988; Sussman et al., 1994; Osher and Fedkiw, 2001; Sethian and Smereka, 2003). In Lagrangian methods (e.g., Unverdi and Tryggvason, 1992; Tryg gvason et al., 2001), particles are typically used to track the movements of the interface. The relative merits of VOF and LS as well as other possibilities are discussed by Lakehal et al. (2002). The next step involves the solution of the unique set of conservation equations over the entire computational domain. Numerical difficulties arise here due to the sharp variation of the fluid properties across the interfaces. A number of methods have been proposed to deal with this problem. In some of these, the sharp variation of the fluid properties is smeared in a narrow band of fluid on both sides of the interface. The second-gradient approach developed by Jamet et al. (2001a, 2001b) is particularly interesting in this respect, since it is based on the thermodynamics of the interface. We will not elaborate on the details of the one-fluid/interface tracking techniques; the interested reader is referred to available reviews (Mc Hyman, 1984; Sethian, 1996; Shyy et al., 1996; Sethian and Smereka, 2003). The ultimate goal, namely to capture the geometry of the interfaces and to resolve sufficiently well the region near the interface and the gradients there, so that heat and mass transfer can be computed, is much more elusive. Interface tracking methods, although not limited in theory to consideration of turbulence in the fluids by methods such as Direct Numerical Simulation (DNS), are in practice not adequate for this. Indeed, the scales needed to consider turbulence are typically orders of magnitude smaller than those used for the resolution of the interfaces in practical problems. The next best alternative would be a combination of Large-Eddy Simulation (LES) with interface tracking methods. Efforts in this direction are underway in our laboratory. It is, however, possible to conduct DNS studies of turbulent flows in certain relatively simple two-phase flow situations, for example, for countercurrent flows of two phases separated by a simple deformable interface. An example of such an application will be discussed in the case study below. This situation also appears in the so-called Pressurized Thermal Shock scenario when emergency coolant (subcooled water) is injected in a pipe containing stratified layers of saturated water and steam and produces rapid condensation but also sudden cooling of the pipe walls; for recent progress in this area, see Yao et al. (2003). 4. Addressing the problems at a multiplicity of scales Sometimes, one attempts to fully understand a situation by considering a cascade of problems at various scales with a corresponding panoply and hierarchy of tools. For the nuclear systems that motivated such developments, the behaviour of the entire system is typically obtained using a system code based on the two-fluid approach and operating at scales comparable to the dimensions of the system and its components. Local phenomena, or the behaviour of parts of the system, may need then to be addressed at the mesoscale level, with tools considering smaller scales and more detailed description of phenomena. Finally, one may need to obtain wall and interfacial momentum, heat

5 G. Yadigaroglu / Nuclear Engineering and Design 235 (2005) Case study: condensation of large bubbles in a pool of water Fig. 2. The computational cascade. and mass transfer laws by performing studies at the smallest possible scale, for example, via DNS of turbulence; such level of spatial resolution is indeed needed to resolve the gradients determining transfers at the interfaces. An example of such an approach will be given in the case study already mentioned. In other words, certain problems may have to be addressed with a cascade of computations. At each level of the scale hierarchy, the physics of the flow may be amenable to numerical prediction by scale-specific strategies. Cross-scale interactions (feed-forward and feedback between micro-, meso-, macro-scales) require merging of the solutions delivered by scalespecific approaches at each level of the scale hierarchy, as sketched in Fig. 2. Such approaches are among the goals of the NURESIM project proposal prepared for execution during the Euratom Sixth Framework Programme (Latrobe et al., 2003). Pushing the computations and the scales considered to the nanoscale level, Molecular Dynamics (e.g., Mouriyama, 2000) is the ultimate tool in CFD and CMFD. We will only mention here, as a relevant example, the possibility of investigating phenomena such as the vaporisation of an ultra-thin liquid layer on a hot metallic surface by Molecular Dynamics simulations (Pan et al., 2002). In this case, the forces acting between all combinations of pairs of wall and fluid molecules are modelled and the evolution of the system is simulated numerically; the results of Pan et al. (2002) showed resemblance to our knowledge of the vaporisation of a similar macroscopic system. The case study discussed below was motivated by the need to understand certain phenomena taking place in passive Boiling Water Reactor (BWR) containments, in particular the containment of the ESBWR (Rao and Gonzalez, 1998). The situation of interest is the condensation of large bubbles injected into a pool of water from a large downwards-pointing vent having a relatively shallow immersion depth. The bubbles contain a mixture of steam and non-condensable gases. From the system (macro-scale) point of view, one is interested in finding out whether there is direct communication between the exit of the vent and the surface of the pool; in this case, condensation will not take place in the pool, something that should be avoided. For the real plant vent diameters and flow rates, experimentation at a scale of 1:1 was too expensive to consider. So, there was a strong incentive to develop and assess computational techniques capable of providing the answer. The phenomena of interest are the growth of the bubble at the vent, its rise and eventual break-up (meso-scale). Predicting bubble break-up is important since, after break-up, the smaller bubbles condense very rapidly. Assessing the rate of condensation is important during the growth and rise phases of the large bubbles, in particular in the presence of non-condensables that degrade the rate of condensation (micro-scale). In this cascade of analyses, the system code provides boundary conditions for the local analyses and the detailed microscopic-level investigations provide the interfacial heat, mass and momentum transfer laws needed to close the problem at the intermediate level. Clearly, the three levels of analysis are coupled and information must be fed back and forth from one level to the other VOF simulations of downwards injection from a vent Meier et al. (2002) produced VOF simulations for the injection of air bubbles into water; these mimicked well experimental findings, in spite of the fact that they were conducted in axisymmetric geometry (Fig. 3). The real situation is only partly axisymmetric; most bubbles, after a roughly axisymmetric initial growth period, tilt to one side or the other and also develop azimuthal instabilities that lead to their break-up. These

6 158 G. Yadigaroglu / Nuclear Engineering and Design 235 (2005) Fig. 3. Bubble formation from injection of air through a downward-facing vent. High-speed video images (bottom) and VOF computations (top) (Meier et al., 2002). details could not of course be simulated with axisymmetric computations, but were reproduced in later 3D work mentioned below. The axisymmetric computations reproduced, however, fairly well the characteristic frequency of appearance of bubbles at the exit of the vent, as well as their size and shape at break-up. Fig. 3 shows a number of frames from both experimental recordings and the corresponding VOF simulations. Electronic Annexes 1 and 2 show a sample experimental recording and an animation from a VOF computation. The small-scale experiment (injection from a 5-cm diameter, downwards facing vent) was set up to verify the computations (Meier, 1999; Meier et al., 2000). Very recently, Liovic (2003) simulated the Meier downwards vent data with a 3D VOF technique (Liovic

7 G. Yadigaroglu / Nuclear Engineering and Design 235 (2005) Fig. 4. Three-dimensional VOF simulation of downwards injection of air showing azimuthal instabilities (experimental recording at the bottom) (Liovic, 2003). Wilson, 2000), Meier et al. (2002) and Liovic (2003) could not compute the behaviour of bubbles containing also steam for lack of a condensation heat and mass transfer law and because of difficulties in the integration of such a law within the VOF computations. The difficulties inherent to the heat and mass transfer physics of the problem can be clearly shown in this case study: for Prandtl and Schmidt numbers typical of the present situation, the thickness of the regions over which the noncondensable gas concentration and liquid and vapour temperature gradients are significant is a fraction of a millimetre (Meier, 1999) and resolution of these boundary layers is not possible at the scale at which the VOF simulations are performed. Thus, heat and mass transfer cannot be computed properly with the resolution of the fields used for the VOF computations. Other ways must be found for determining the interfacial exchange laws and incorporating them into the VOF simulation. Measurements of the instantaneous rate of condensation at the surface of large bubbles are difficult and available experimental data refer to rather simple situations like the condensation of small spherical bubbles. In the absence of experimental information specific to the problem in hand, one can try to apply correlations derived from the solution of simple problems approximating the situation locally at the interface of the large condensing bubble. Davis and Yadigaroglu (2004) solved with a combination of classical analytical techniques augmented with numerical computations the problem of condensation of pure steam impinging on a liquid surface and creating a stagnation-flow situation. Fig. 5 shows how the solution to this problem may approximate the more complex reality. Davis et al., 2002). Fig. 4 shows a frame from his computations. The surface instabilities that create the azimuthal ripples present in the experimental recordings are clearly visible now. The 3D computations require, however, much greater computing power and time Heat and mass transfer at the condensing interface The simulation of the condensation of the steam contained in the large bubbles requires the implementation of heat and mass transfer capability in the VOF code used. Although this is possible (e.g., Welch and Fig. 5. The direct-contact condensing flows considered by Davis and Yadigaroglu (2004) is shown on the right. The vent flow situation that should be simulated is shown on the left.

8 160 G. Yadigaroglu / Nuclear Engineering and Design 235 (2005) and Yadigaroglu covered a wide range of parameters and could correlate their results for further use. Gerner and Tien (1989) presented a similar but axisymmetric solution for two impinging stagnation jets over a flat interface which, however, also took into account non-condensables. The remaining challenge is to link such correlations to the computed flow conditions on both sides of the interface. Indeed, the theoretical solution is given in terms of the potential flow velocity outside the boundary layer created by the impinging flow. One has to link via an appropriate algorithm the computed flow conditions on both sides of the interface (in the liquid and the gas, at some distance from the interface) to the idealized, corresponding, theoretical potential flow velocities DNS of turbulent condensing flow As the last resort, one can rely on DNS of turbulence in the flow to elucidate the fundamental laws of condensation in the presence of non-condensables. DNS computations can be performed in an idealised configuration, in which steam or a steam/air mixture flow counter-currently over a liquid surface in a rectangular box (Fig. 6). The steam condenses on the interface. Lakehal et al. (2003) and Fulgosi (2004) performed DNSs of condensation of pure, saturated or superheated steam on subcooled water in a stratified, counter-current flow situation. First one needs to compute and describe the flow and turbulence structure in the thin diffusive layers on either side of the interface; this is essentially an extension of earlier work per- Fig. 6. The computational domain used for the DNS of countercurrent flow of steam and water (top) and variation of the main variables near the interface (bottom).

9 G. Yadigaroglu / Nuclear Engineering and Design 235 (2005) formed without condensation by Lombardi et al. (1996). Next, the condensation heat transfer laws are obtained from the DNS data (Lakehal et al., 2003). The final steps towards obtaining the interfacial exchange laws in the presence of noncondensables are underway. In the work of Fulgosi and co-workers mentioned here, the DNS computations are conducted separately in each phase and are coupled with jump conditions considering the interfacial exchanges at the liquid vapour interface. The latter is deformable to a certain extent (small ripples can be accommodated, but no large breaking waves) and its position is continuously tracked. The boundary layers on both sides of the interface grow in the computational box that has periodic boundary conditions over its bounding vertical planes. Therefore, special care should be taken to maintain the correct boundary conditions at the upper and lower horizontal surfaces of the box; the condensation mass flux at the interface should be added and extracted at these boundaries, respectively, to produce steadystate data from which the heat and mass transfer laws can be extracted. Care should also be taken to properly simulate the turbulence balances at the boundaries. The results obtained so far agree with available experimental data. The work of Fulgosi (2004) provided firstof-a-kind detailed data and produced insights into the effects of condensation on turbulence in the boundary layers. 6. Multi-scale critical heat flux predictions with CMFD? The modelling and computation of the Critical Heat Flux (CHF) condition has been a very elusive target for the last several decades. In spite of the enormous number of publications on the subject, there is still no universal model and tool for reliable CHF predictions under all conditions. At the European Two-Phase Flow Group meeting that took place in Stockholm in June 2002, our French colleagues (Bestion, 2002) proposed the creation of a CMFD initiative aiming at the prediction of the CHF condition; this idea is expanded here. Both subcooled and low-quality Departure from Nucleate Boiling (DNB), and high-quality dryout (DO) could be attacked with a panoply of CMFD tools and cascades of computations at micro-, meso-, and macroscales. For DNB at low quality, the microscale work would involve bubble nucleation at the wall (e.g., VOF or LS with heat and mass transfer); such computations are at their infancy, but work in this direction has already been published. For example, Welch (1998), Son et al. (2002) and Mathieu et al. (2004) produced computations of bubble growth on a heated wall. The mesoscale computations would deal with the bubbly layer near the wall; bubbly flows have been already treated with the interpenetrating-media approach, including (so far rather tentative) RANS models of turbulence (Lance et al., 1999). More advanced approaches, for example, with LES are being attempted (Milelli et al., 2001); Dean et al. (2004) present interesting recent developments in this area. Finally at the macroscale, the entire flow channel would be considered, including the interaction of the wall layer with the bulk fluid. For dryout at high quality, the microscale computations would again address the bubbles and nucleation at the wall; the generation and detachment of waves from the liquid film; the impingement of drops and their capture on the liquid film; heat and mass transfer from the surface of the film and from the wall. VOF, LS and DNS could again be the candidate techniques for such computations. At the mesoscale, the global liquid film mass balance and heat and mass transfer from/to the liquid film should be considered. Finally, at the macroscale, the overall channel condition should be examined and the mesoscale results integrated along the channel to arrive at the DO condition. Such an approach is easier to describe than to actually implement. Although the exercise will not lead to full success soon, the trip will certainly be worth the effort in terms of fallout and spin-off developments. 7. Conclusions Although CFD of single-phase flows has reached a certain degree of maturity, a number of Computational Multi-Fluid Dynamics (CMFD) methods are still under development. Various situations or phenomena can be best addressed at a multiplicity of time/space scales: the micro-, meso-, macro-scales. At each level of the scale hierarchy, the physics of the flow may be best amenable to numerical prediction by scale-specific strategies. Cross-scale interactions (feed-forward and -

10 162 G. Yadigaroglu / Nuclear Engineering and Design 235 (2005) back between scales) require merging of the solutions delivered by the scale-specific approaches. Clearly, the way to such multi-scale treatments requires first advances in the corresponding scale-specific methods. The case study outlined in this paper and the CHF initiative mentioned illustrate what we have referred to as cascades of CMFD methods. To deal with the venting of steam/air mixtures from a vertical downward vent, we had to address the problem at various scales with a variety of tools: system behaviour (in the example cited here, the flow rate and the composition of the mixture entering the vent) with a system code; large bubble behaviour with VOF; finally, the direct-contact condensation heat transfer law via DNS and analytical methods. Cascades of computations at different scales would also be needed to arrive at the grand-challenge, the CMFD of CHF. A number of interesting international collaborative developments are taking place to develop a new generation of tools for safety analysis. Several projects such as ASTAR and ECORA were conducted in Europe under the fourth and fifth FWP (FISA, 2001, 2003) and the effort is likely to culminate in the sixth FWP, the NURESIM project under preparation. As illustrated by the examples discussed in this paper, in the future, safety issues are more likely to be addressed at a variety of scales with a panoply of (partly) new tools and methods, including CMFD. Next to the classical, mature system codes, there is indeed also room for problem-specific computing platforms or codes, working at a multiplicity of scales, when necessary considering multiple physics, and coupled to the classical system analysis codes. Acknowledgments The author is gratefully acknowledging the contributions of his ETH collaborators who performed most of the computations presented in this paper: D. Lakehal, J. Davis, M. Fulgosi, P. Liovic, and M. Meier. The Electronic Annexes were produced by M. Meier. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi: /j. nucengdes References Bestion, D., Capabilities and limitations of thermal hydraulic codes. In: Réocreux, M., Rubinstein, M.C. (Eds.), Proceedings of the CSNI Specialist Meeting on Transient Two- Phase Flow, Aix-en-Provence, 6 8 April 1992, pp Bestion, D., Oral communication at European Two-Phase Flow Group Meeting, Stockholm, June Carlson, K.E., et al., RELAP5/MOD3 code manual, vol. I, Code structure, system models, and solution methods, NUREG/CR-5535, EGG-2596 (June). Damerell, P.S., Simons, J.W., D/3D program, Work summary report, and reactor safety issues resolved by the 2D/3D Program, US NRC Reports NUREG/IA-0126 and -0127, GRS-100 and -101, MPR-1345 and (June). Davis, J., Yadigaroglu, G., Direct contact condensation in Hiemenz flow boundary layers. Int. J. Heat Mass Transfer 47, Dean, G.N., van den Hengel, E.I.V., van Sint Annaland, M., Kuipers, J.A.M., Multi-scale modeling of dispersed gas liquid twophase flows. In: Proceedings of the 5th International Conference on Multiphase Flow, Keynote Lecture K-07, ICMF-2004, Yokohama, Japan, 30 May 4 June. EG&G Idaho Inc., RELAP4/MOD6, A computer program for transient thermal hydraulic analysis of nuclear reactors and related systems, User s Manual, CDAP TR 003. FISA, In: Proceedings of the EU Research in Reactor Safety, Luxembourg, EURATOM, Brussels (EUR 20281), November FISA, In: Proceedings of the EU Research in Reactor Safety, EC Luxembourg, November Fulgosi, M., DNS of turbulent heat transfer and condensation in stratified flow with a deformable interface, Doctoral dissertation no Swiss Federal Institute of Technology Zurich. Gerner, F., Tien, C., Axi-symmetric interfacial condensation model. J. Heat Transfer 111, Hirt, C.W., Nichols, B.D., Volume of Fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys. 39, 201. Ishii, M., Two-fluid model based on interfacial area transport equation. In: Proceedings of the 5th International Conference on Multiphase Flow, Plenary Lecture 4, ICMF-2004, Yokohama, Japan, 30 May 4 June. Jamet, D., Lebaigue, O., Coutris, N., Delhaye, J.-M., 2001a. The second gradient theory: a tool for the direct numerical simulation of liquid vapor flows with phase-change. Nucl. Eng. Des. 204, Jamet, D., Lebaigue, O., Coutris, N., Delhaye, J.-M., 2001b. The second gradient method for the direct numerical simulation of liquid vapor flows with phase-change. J. Comput. Phys. 169, Lakehal, D., Meier, M., Fulgosi, M., Interface tracking towards the direct simulation of heat and mass transfer in multiphase flows. Int. J. Heat Fluid Flow 23, Lakehal, D., Fulgosi, M., Yadigaroglu, G., Banerjee, S., Direct Numerical Simulation of turbulent heat transfer across a mo-

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