COMPUTATIONAL FLUID DYNAMICS FOR NUCLEAR APPLICATIONS: FROM CFD TO MULTI-SCALE CMFD

Transcription

1 COMPUTATIONAL FLUID DYNAMICS FOR NUCLEAR APPLICATIONS: FROM CFD TO MULTI-SCALE CMFD G. Yadigaroglu Swiss Federal Institute of Technology (ETH), Nuclear Engineering Laboratory, Zurich (CH) SUMMARY Although CFD of single phase flows has been commonplace, Computational Multi- Fluid Dynamics (CMFD) is still under development. For certain problems, one may have to conduct cascades of computations with 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 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 was treated by Direct Numerical Simulation (DNS). A. INTRODUCTION A paper dealing with the trends in numerical simulation for Light Water Reactor (LWR) safety was presented at the FISA-2001 conference [1]. That paper was essentially limited to the application of Computational Fluid Dynamics (CFD) methods to singlephase 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 (CMFD) [2] applications. We also discuss and illustrate with a case study the approach consisting of addressing complex multi-phase flow problems involving a range of space and time scales with cascades of computations at different scales now the multi-scale approach [3]. Multiphase system and component analysis codes are indispensable in reactor safety analysis. The very first system codes were based on the equal-velocities, equal-phasetemperatures, homogeneous-equilibrium model. Separated-flow models, where the two phases are allowed to have different average velocities followed; in this case, the averagevelocity ratio is typically derived from some empirical correlation. The drift-flux model, characterizing this velocity ratio with two parameters having a clear physical significance 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 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 model was introduced to deal with these situations; it is widely used today. This is the level of development currently implemented and used in most system codes. Although the development of three-dimensional (3D) multiphase simulation tools for safety analysis and reactor design 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.

2 B. TWO-FLUID VERSUS ONE-FLUID, INTERFACIAL-TRACKING FORMULATIONS It is worth recalling the basic premise of the two-fluid model at this point. The twofluid phase 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, considering the presence of the interfaces during the local averaging process. This leads to the definition of the local interfacial area concentration that provides the area for the exchanges of mass, momentum and energy between the phases. However, 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, at a say 30% void fraction, the flow is stratified or bubbly. This is fine with many problems, but there are situations where the two phases are sharply separated (at a large scale, such as the scale of the duct) and full understanding of the situation requires knowledge of the position 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 some way, 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 is necessary to get solutions for such problems. There are also cases where prediction of the location 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. C. ONE-FLUID FORMULATION AND INTERFACE TRACKING METHODS The various interface tracking methods are typically associated with a one-fluid description of the two-phase flow. In this formulation, unique conservation equations are 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 position of the interface is tracked using a variety of procedures [4]. Interface tracking methods can be Lagrangian or Eulerian. The most frequently employed Eulerian interface tracking methods are the Volume of Fluid (VOF) method and the Level Set (LS) method. The relative merits of VOF and LS, as well as other possibilities are discussed by Lakehal et al. [4].

3 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 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 [5]. It is, however, possible to conduct DNS studies in certain relatively simple twophase flow situations, for example, for countercurrent flows of two phases separated by a simple interface; the latter can be deformable within limits; an example of such an application will be discussed in the case study below. D. ADDRESSING THE PROBLEMS AT A MULTIPLICITY OF SCALES Some times, 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 this work, 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 and mass transfer laws by performing studies at the smallest possible scale, for example, via DNS; 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 scale-specific approaches at each level of the scale hierarchy. Such approaches are among the goals of the NURESIM project proposal prepared for execution during the Euratom 6th Framework Programme (FWP). Pushing the computations and the scales considered to the nanoscale level, Molecular Dynamics 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 ultrathin liquid layer on a hot metallic surface by Molecular Dynamics simulations [6]. 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 Ref. [6] showed resemblance to our knowledge of the vaporisation of a similar macroscopic system. E. CASE STUDY: CONDENSATION OF LARGE BUBBLES IN A POOL OF WATER The case study discussed below was motivated by the need to understand certain phenomena taking place in passive Boiling Water Reactor (BWR) containments, namely the condensation of large bubbles injected into a pool of water from a large downwards-

4 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, in particular in the presence of non-condensables that degrade the rate of condensation (micro-scale). In this cascade of analyses, the system code will provide boundary conditions for the local analyses and the detailed microscopic-level investigations will provide the interfacial heat, mass and momentum transfer laws needed to close the problem at the intermediate level. F. VOF SIMULATIONS OF DOWNWARDS INJECTION FROM A VENT Meier et al. [7] 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 (Figure 1). 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 details could not of course be simulated with axisymmetric computations, but were reproduced in later 3D work outlined 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. Figure 1 shows a number of frames from both experimental recordings and the corresponding VOF simulations. The small-scale experiment (injection from a 5-cm diameter, downwards facing vent) was set up to verify the computations. Very recently, Liovic [8] simulated the Meier downwards vent data with a 3D VOF technique [9]. Figure 2 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. G. HEAT AND MASS TRANSFER AT THE CONDENSING INTERFACE Meier [7] and Liovic [8] could not compute the behaviour of bubbles containing also steam for lack of a condensation heat and mass transfer law. 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 [10] and resolution of these boundary layers is not possible at the scale at which the VOF simulations are performed. Other ways must be found for determining the interfacial exchange laws and incorporating them into the VOF simulation.

5 Figure 1: Bubble formation from injection of air through a downward facing vent. High-speed video images (bottom) and VOF computations ( top) [7]. Figure 2: Three-dimensional VOF simulation of downwards injection of air showing azimuthal instabilities (experimental recording at the bottom) [8]. In the absence of experimental information, one can try to apply correlations derived from the solution of simple problems. Davis and Yadigaroglu [11] solved with a combination of classical analytical techniques augmented with numerical computations the problem of condensation of pure steam impinging on a liquid surface, Figure 3. They covered a wide range of parameters and could correlate their results for further use. The remaining challenge is to link such correlations to computed reference conditions on both sides of the interface. Figure 3: Direct-contact condensing flows considered by Davis and Yadigaroglu [11], right. The vent flow situation that should be simulated is shown on the left.

6 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, Figure 4. The steam condenses on the interface. One has to compute and describe the flow and turbulence structure in the thin diffusive layers on either side of the interface (Figure 4) as an extension of earlier work without condensation [12]. The final steps towards obtaining the interfacial exchange laws with and without noncondensables are underway as a doctoral dissertation at ETH [13-14]. 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). 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 achieve a steady state 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. Figure 4: The computational domain used for the DNS of countercurrent flow of steam and water (left) and variation of the main variables near the interface (right). H. 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 [15] 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 highquality dryout (DO) could be attacked with a panoply of CMFD tools and cascades of computations at micro-, meso-, and macro-scales.

7 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 have already been published. 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 [16]. More advanced approaches, for example, with LES are being attempted [17]. 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 success soon, the trip will certainly be worth the effort in terms of fallout and spin-off developments. G. 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. The case study outlined above and the CHF initiative illustrate what we have referred to as cascades of CMFD methods. To deal with the venting of the steam/air mixtures, 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 directcontact condensation heat transfer law via DNS. 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 such projects were conducted in Europe under the 4th and 5th FWP and the effort is likely to culminate in the 6th FWP. 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. 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.

8 REFERENCES [1] G. Yadigaroglu, M. Andreani, J. Dreier and P. Coddington, Trends and needs in experimentation and numerical simulation for LWR safety, Nucl. Engng and Design, 221 pp (2003). [2] G. Yadigaroglu, Letter to the Editor, CMFD (a brand name) and other acronyms, Int. J. Multiphase Flow, 29, pp (2003). [3] G. Yadigaroglu, Application of cascades of CFD methods to two-phase flow problems, Trans. Inst. of Fluid-Flow Machinery (Poland), 112, pp (2003). [4] D. Lakehal, M. Meier and M. Fulgosi, Interface tracking towards the direct simulation of heat and mass transfer in multiphase flows, Int. J. of Heat and Fluid Flow, 23, pp (2002). [5] D. Lakehal and P. Liovic, private communication (2003). [6] Pan Yi, D. Poulikakos, J. Walther and G. Yadigaroglu, Molecular Dynamics simulation of vaporization of an ultra-thin liquid argon layer on a surface, Int. J. Heat and Mass Transfer, 45, pp (2002). [7] M. Meier, G. Yadigaroglu and B.L. Smith, A novel technique for including surface tension in PLIC-VOF methods, Eur. J. Mech. B/Fluids, 21, pp (2002). [8] P. Liovic, private communication, work in progress (2003). [9] P. Liovic, M. Rudman and J.-L. Liow, Numerical modelling of free surface flows in metallurgical vessels, Applied Mathematical Modelling, 26, pp (2002). [10] M. Meier, Numerical and experimental study of large steam air bubbles injected in a water pool, doctoral dissertation, ETH Zurich, Diss. no (1999). [11] J. Davis and G. Yadigaroglu, Direct contact condensation in Falkner-Skan flows, Proceedings of IMECE 02, 2002 ASME International Mechanical Engineering Congress & Exposition, November 17-22, 2002, New Orleans, Louisiana, USA, paper IMECE [12] P. Lombardi, V. De Angelis and S. Banerjee, Direct numerical simulation of nearinterface turbulence in coupled gas-liquid flow, Phys. Fluids, 8, pp (1996). [13] M. Fulgosi, D. Lakehal, S. Banerjee and G. Yadigaroglu, Direct Numerical Simulation of turbulence and interfacial dynamics in counter-current air-water flows, pp in Proceedings DLES-4, Direct and Large-Eddy Simulation IV, ERCOFTAC Series, Vol. 8, B.J. Geurts, R. Friedrich, O. Métais (Eds.), Kluwer Academic Publishers, Dordrecht (2001). [14] M. Fulgosi, D. Lakehal, S. Banerjee and V. De Angelis, Direct Numerical Simulation of turbulence in a sheared air-water flow with a deformable interface, J. Fluid Mechanics, 482, pp (2003). [15] D. Bestion, oral communication at European Two-Phase Flow Group Meeting, Stockholm, June [16] M. Lance, J. L. Marié and J. Bataille, Turbulence in bubbly flows: from experiments to numerical modelling, pp in Two-Phase Flow Modelling and Experimentation 1999, G.P. Gelata, P. Di Marco and R.K. Shah (editors), Edizioni ETS, Pisa. [17] M. Milelli, B.L. Smith and D. Lakehal, Large Eddy Simulation of turbulent shear flows laden with bubbles, pp in Proceedings DLES-4, Direct and Large- Eddy Simulation IV, ERCOFTAC Series, Vol. 8, B.J. Geurts, R. Friedrich, O. Métais (Eds.), Kluwer Academic Publishers, Dordrecht (2001).