Basic Concepts about CFD Models

Lappeenranta University of Technology Summer School in Heat and Mass Transfer August 18 20, 2010 Basic Concepts about CFD Models Walter Ambrosini Associate Professor in Nuclear Plants at the University of Pisa 1

Summary General remarks on turbulent flow Instability of laminar flow Statistical treatment of turbulent flow Momentum transfer in turbulent flow Heat transfer in turbulent flow Basic concepts about computational modelling of turbulent flows Length scales in turbulence Direct Numerical Simulation (DNS) Large Eddy Simulation (LES) Reynolds Averaged Navier-Stokes equations (RANS) Two-phase flow applications Prediction of heat transfer deterioration 2

General remarks on turbulent flow Instability of Laminar Flow - 1 The transition from laminar flow to turbulence is an example of flow instability: beyond a certain threshold, inertia overcomes viscous forces and the motion cannot be anymore ordered this was shown by Osborne Reynolds in a classical experiment 3

General remarks on turbulent flow Instability of Laminar Flow - 2 This transition occurs in many different systems: pipe flow boundary layers 4

General remarks on turbulent flow Instability of Laminar Flow - 3 free jets wakes 5

General remarks on turbulent flow Instability of Laminar Flow - 4 In order to study stability of a nonlinear system by analytical means the methodology of linear stability analysis is often adopted This has the objective to determine the stability conditions consequent to infinitesimal perturbations: : e.g., for a 2D boundary layer it is EXAMPLES OF TRANSIENT ANALYSES Cavity RB Convection Buoyant Jet 6

General remarks on turbulent flow Instability of Laminar Flow - 5 Turbulence introduces a large degree of sensitivity to initial conditions (SIC) that is typical of deterministic chaos By this, it is meant that turbulent motion is not random, though it appears fluctuating in a similar manner, since the equations governing the system are well defined This characteristic is shared with many different chaotic systems,, even governed by simple equations dre dτ = Gr Ψ 1 2 - L f'(re) Re Re D dψ 1 dτ = π Re Ω 1 - π2 Fo Ψ 1 + 4 π sin γ Cooling γ dω 1 dτ = - π Re Ψ 1 - π2 Fo Ω 1 + 4 π cos γ Heating 7

General remarks on turbulent flow Statistical Treatment of Turbulent Flow - 1 Owing to the fluctuating nature of the turbulent flow field, it is customary (after Reynolds) to introduce an appropriate time averaging of any specific value ( intensive( intensive ) ) of major extensive variables The attempt is quite evidently to write equations in terms of time averaged variables,, structurally similar to those of laminar flow 8 This attempt is successful, but fluctuations cannot be forgotten

General remarks on turbulent flow Statistical Treatment of Turbulent Flow - 2 In particular, the following quantities have overwhelming importance Turbulence intensity is strictly related to the turbulence kinetic ic energy This is one of the most important quantities adopted in present CFD codes,, mostly making use of two-equation models,, to be described later on 9

General remarks on turbulent flow Statistical Treatment of Turbulent Flow - 3 The general balance equations in local and instantaneous formulation are then averaged making use of the above described averaging operator After simplifications (described in lecture notes), an averaged form is finally reached showing that the attempt to get equations similar to those of laminar flow leaves an additional term This term, having a clear advective nature, points out that fluctuations do play a role in transfers: this role represents a 10 sort of additional mixing due to turbulence

General remarks on turbulent flow Statistical Treatment of Turbulent Flow - 4 In analogy with the molecular motion, the basic idea is therefore to interpret such term as an additional diffusion due to turbulence The momentum and energy balance equations contain this term that calls for a proper modelling 11

General remarks on turbulent flow Momentum Transfer in Turbulent Flow - 1 The Reynolds stress tensor appears in momentum equations The Reynolds stresses account for the additional momentum flux due to eddies 12

General remarks on turbulent flow Momentum Transfer in Turbulent Flow - 2 It is then customary to adopt the Boussinesq approximation based on a definition of turbulent momentum diffusivity (eddy viscosity),, trying to define a simple constitutive relationship for the Reynolds stress The quantity ν T is no more a property of the fluid, but also depends on flow. Of course, the Boussinesq approximation shifts the toughness of the modelling problem to the definition of the eddy viscosity 13 of the modelling problem to the definition of the eddy viscosity

General remarks on turbulent flow Momentum Transfer in Turbulent Flow - 3 By the way, many different kinds of turbulence can be envisaged, ranging from ideally homogeneous and isotropic to more realistically heterogeneous and anisotropic Wall turbulence is a classical example of the latter cases: Eddy viscosity models have therefore the very tough job to reintroduce the complexity lost in the simple Boussinesq 14 approximation

General remarks on turbulent flow Momentum Transfer in Turbulent Flow - 4 It is rather instructive and useful to consider the distribution of velocity close to a plane wall; ; different quantities of widespread use in CFD are introduced at this stage A universal logarithmic velocity profile is found both on the basis of simple theoretical considerations and experiments 15

General remarks on turbulent flow Momentum Transfer in Turbulent Flow - 5 The effect of turbulence in the transport of momentum can be clearly seen in comparing the distributions of velocity in the classical case of a circular pipe for laminar and turbulent flows The flatter profile observed in the case of turbulent flow is the direct consequence of the increasing efficiency in momentum transfer far from the wall due to the mixing promoted by 16 turbulence

General remarks on turbulent flow Heat Transfer in Turbulent Flow - 1 The averaged total energy equation and the steady thermal energy equation in terms of temperature can be written as Also in these cases additional terms to be modelled appear, e.g.: The rationale for evaluating the turbulent contribution is similar as in the case of momentum where α T is the turbulent thermal diffusivity 17

General remarks on turbulent flow Heat Transfer in Turbulent Flow - 2 The picture of the turbulent transfer phenomenon is therefore the same as for momentum: The relation between the two turbulent diffusivities of heat and momentum poses an additional problem 18

General remarks on turbulent flow Heat Transfer in Turbulent Flow - 3 A simple but effective way to establish this relationship is to define a constant turbulent Prandtl number, in analogy with the molecular one assuming that, as a consequence of the Reynolds analogy, this could be in the range of unity The assumption in this case is that the same coherent structures carrying momentum are also responsible of heat transfer However, this assumption holds acceptably for fluids having nearly unity molecular Prandtl number; ; in the other cases, different approaches should be used 19

Basic concepts about computational modelling of turbulent flows Length Scales in Turbulence - 1 In turbulent flow an energy cascade occurs representing the transfer of turbulence kinetic energy from larger to smaller eddies As such, turbulence can be considered as a phenomenon characterised by a wide range of lengths at which interesting phenomena do occur: from the integral length scale, l,, at which energy is extracted from the mean flow to the Kolmogorov length scale, η,, at which turbulence kinetic energy is finally dissipated into heat 20

Basic concepts about computational modelling of turbulent flows Length Scales in Turbulence - 2 It must be noted that the Kolmogorov length scale, η, is small but still large with respect to the molecular mean free path : so, turbulence can still be studied basing on the continuum assumption The integral length scale, l, characterising large eddies can be defined as the average length over which a fluctuating component keeps correlated, i.e. the quantity is not negligible On both dimensional and experimental basis, it can be shown that and with ; therefore, 21

Basic concepts about computational modelling of turbulent flows Length Scales in Turbulence - 3 Basing on these considerations, it can be concluded that: an adequate representation of turbulence should take into account the phenomena of production and dissipation of turbulence kinetic energy at the different scales in this respect, two different strategies can be envisaged: simulating the transient evolution of vortices of different sizes,, putting a convenient lower bound for the smallest scale (DNS, LES, DES) simulating turbulence on the basis of the above described statistical approach,, introducing appropriate production and dissipation terms to approximately represent the effects of the energy cascade (RANS) 22

Basic concepts about computational modelling of turbulent flows Direct Numerical Simulation (DNS) - 1 This methodology follows the former of the two mentioned routes, trying to simulate with the highest possible space and time detail the evolution of vortices of all relevant sizes The assumption behind this technique is that the Navier-Stokes equations are rich enough to describe the turbulent flow behaviour with no need of additional constitutive laws; for incompressible flow it is: The web is full of fascinating pictures and movies about DNS results 23

Basic concepts about computational modelling of turbulent flows Direct Numerical Simulation (DNS) - 2 The application of this technique is very demanding in terms of computational resources: : representing flows of technical interest is very challenging and requires massive parallel computing However the technique is very promising and it is sometimes used to provide data having a similar reliability to experiments with greater detail in local values In fact, if used with enough detail, DNS can provide data which can be hardly obtained in similar detail with experiments In addition to be an interesting field of research, DNS is therefore used also to provide data on which empirical turbulence model can be validated 24 CFD-Figure Figure-1.ppt

Basic concepts about computational modelling of turbulent flows Large Eddy Simulation (LES) - 1 At a more reduced level of detail, LES is aimed at simulating only larger eddies, while the smaller scales are treated by subgrid-scale scale (SGS) models In other words, there are two different length scales: the large scales that are directly solved as in DNS; the smaller scales that are treated by SGS models As such, LES is computationally more efficient than DNS and may be also relatively accurate A key point in LES is introducing a spatial filtering for the smaller scales 25

Basic concepts about computational modelling of turbulent flows Large Eddy Simulation (LES) - 2 The filters can be of different types: 26

Basic concepts about computational modelling of turbulent flows Large Eddy Simulation (LES) - 3 27

Basic concepts about computational modelling of turbulent flows Large Eddy Simulation (LES) - 4 Once the resolvable scales are defined, the averaged N-S N S equations are written in averaged form 28

Basic concepts about computational modelling of turbulent flows Large Eddy Simulation (LES) - 5 The advection term can be manipulated as or also Anyway, introducing the subgrid-scale scale stresses (or adopting slightly different definitions) it can be finally obtained 29

Basic concepts about computational modelling of turbulent flows Large Eddy Simulation (LES) - 6 So, the fundamental problem is defining the subgrid scale stresses In 1963, Smagorinsky defined a model based on the following equations where C S is the Smagorinsky coefficient representing a parameter to be adjusted for the particular problem to be dealt with; values in the range 0.10 to 0.24 have been adopted for typical problems LES is presently promising as a design tool, but still heavy from the 30 computational point of view

Basic concepts about computational modelling of turbulent flows Reynolds Averaged Navier-Stokes (RANS) models - 1 As already mentioned, the Reynolds averaging process leads to momentum equations in which turbulence is represented by the Reynolds stress The Boussinesq approximation suggests that τ 2 2 2 ρν w w i j ij T Sij ij T ij 3 ρ k δ ρν = = + k x j x i 3 ρ δ Moreover if the Reynolds analogy is adopted by specifying a constant turbulent Prandtl number, also the eddy thermal diffusivity is related to the eddy viscosity 31 So, the main problem is reduced to specifying the eddy viscosity

Basic concepts about computational modelling of turbulent flows Reynolds Averaged Navier-Stokes (RANS) models - 2 Models of different complexity can be adopted in this aim, classified on the basis of the number of the additional partial differential equations to be solved: 1. Algebraic or zero-equation equation models 2. One-equation equation models 3. Two-equation models An important distinction between turbulence models is anyway the one between complete and incomplete models: completeness of the model is related to its capability to automatically define a characteristic length of turbulence in a complete model, therefore, only the initial and boundary conditions are specified,, with no need to define case by case parameters depending on the particular considered flow 32

Basic concepts about computational modelling of turbulent flows Reynolds Averaged Navier-Stokes (RANS) models - 3 ALGEBRAIC MODELS Possibly the best known algebraic model is the one obtained by the t mixing length theory of Prandtl (1925) where l mix is the mixing length; the model is similar to the one for molecular viscosity in which kinematic viscosity is a interpreted as the product of a mean molecular velocity by a length (the mean free f path) In the presence of a wall, it is assumed where the constant must be adjusted on an empirical basis The mixing length theory has received different reformulations, but its character of incompleteness makes models based on transport equations to be preferable 33

Basic concepts about computational modelling of turbulent flows Reynolds Averaged Navier-Stokes (RANS) models - 4 PARTIAL DIFFERENTIAL EQUATION MODELS Referring from here on to the specific Reynolds stress tensor it is possible to derive a Reynolds stress transport model by applying the time averaging operator as follows where it is found 34

Basic concepts about computational modelling of turbulent flows Reynolds Averaged Navier-Stokes (RANS) models - 5 This equation shows the typical difficulties encountered when trying to close the turbulence equations.. In fact: the application of the time-averaging operator to the Navier- Stokes equations makes the Reynolds stress tensor to appear as a SECOND ORDER tensor of correlation between two fluctuating velocity components the derivation of transport equations for the Reynolds stress tensor makes HIGHER ORDER correlation terms to appear The transport equation for turbulent kinetic energy can be obtained by taking the trace of the system of Reynolds stress transport equations; in fact 35

Basic concepts about computational modelling of turbulent flows Reynolds Averaged Navier-Stokes (RANS) models - 6 The k equation has the form The Reynolds stress appearing in this equation has the form and the dissipation term has the form and is evaluated by the relationship 36

Basic concepts about computational modelling of turbulent flows Reynolds Averaged Navier-Stokes (RANS) models - 7 A one equation model was proposed by Prandtl in the form with the additional closure equation In general, one-equation equation models are incomplete, since the turbulence length scale, l, must be defined on a case by case basis; complete versions are anyway available which specify independently this length (e.g., Baldwin- Barth, 1990). In order to obtain complete models, an additional quantity must be 37 defined also subjected to a transport equation

Basic concepts about computational modelling of turbulent flows Reynolds Averaged Navier-Stokes (RANS) models - 8 Two-equation models are mostly based on the definition of this further quantity in the form of ε or ω basing on the following relationships that close the problem (other versions are available) for k-ω models it is in particular for the Wilcox (1998) model it is with appropriate values of the constants and, in particular: 38

Basic concepts about computational modelling of turbulent flows Reynolds Averaged Navier-Stokes (RANS) models - 9 for k-ε models it is the dissipation equation can be derived exactly and has the classical form The standard k-ε model adopts the definitions 39

Basic concepts about computational modelling of turbulent flows Reynolds Averaged Navier-Stokes (RANS) models - 10 As presented, the above turbulence models are mostly suited for dealing with turbulence conditions far from walls When wall phenomena must be dealt with two possible approaches are available: use of wall functions : the logarithmic trend observed for velocity close to a flat surface is assumed to hold approximately near the specific considered wall, together with a corresponding temperature trend; in this case, the value of y+ in the first node close to the wall must be conveniently large (e.g., y+ > 30); use of low Reynolds number models: these models are conceived to simulate the actual trend of turbulence close to the wall, by the adoption of damping functions; the value of y+ in the first node must be very small (typically y+<1) 40

Basic concepts about computational modelling of turbulent flows Reynolds Averaged Navier-Stokes (RANS) models - 11 On one hand, the use of wall functions is computationally convenient,, since refining the mesh close to the wall is expensive in terms of resources (see the figure from Sharabi,, 2008) (a) Wall functions mesh (b) Low-Reynolds number mesh On the other hand, wall functions are not able to properly detect some boundary layer phenomena for which they were not conceived (e.g., buoyancy effects in heat transfer, etc.) Nevertheless, even low-reynolds number models are not always 41 completely accurate

Basic concepts about computational modelling of turbulent flows Damping functions in low-re models In low-reynolds number models the definition of eddy viscosity is changed from the classical formulation to various forms including damping functions, f µ ν T 2 = Cµ fµ k ε f µ 0 for y 0 that provide for the decrease of the eddy viscosity while approaching the wall This allows integration of the turbulence models through the boundary layer up to the wall itself Different assumptions lead to various formulations of the low-re 42 models and, generally, to different results

Basic concepts about computational modelling of turbulent flows Low-Re models vs. wall functions Providing an answer to the question if the use of wall functions should be preferred or not to models having a low-re capability is not trivial, since: it heavily depends on the application it is strictly linked to the purpose of the analysis In this lecture I will propose a case in which WFs are not applicable, since they completely overlook phenomena related to buoyancy In a lecture to come on condensation, I will show that the use of some minimum low-re number capabilities is useful to get relatively good agreement with experimental data though approximate method are also acceptable; ; however, pending questions are: could we afford describing a whole nuclear reactor containment with such a strong refinement at the walls? couldn t t we instead accept a more approximate view of local 43 phenomena to get a reasonable overall picture?

Basic concepts about computational modelling of turbulent flows Anisotropic RANS - 1 The assumption of an isotropic value of ν T is not suitable for simulating details of flow in noncircular passages This is particularly true for secondary flows in the direction orthogonal to the main flow that would require the full Reynolds stress transport models to be predicted This choice is anyway heavy for the number of equations to be solved RSM application from Sharabi (2008) A further possibility is to use an anisotropic RANS models in which the simple Boussinesq approximation is abandoned 44

Basic concepts about computational modelling of turbulent flows Anisotropic RANS - 2 In particular, it is possible to use algebraic expressions of the kind (see e.g., Baglietto et al., 2006) which is limited to second order terms in the strain and the rotational rates S ij and Ω ij with respect to the original third order formulation (Baglietto et al., 2006) 45

Two-phase flow applications Few general considerations Two-phase flow introduces additional complexity to the already complex problem of simulating turbulent flow The presence of two phases and of the related interfaces requires particular care in modelling Ambitious goals of modelling two-phase flow with CFD would be, for instance, to represent important phenomena like CHF from first principles 46

Two-phase flow applications Few general considerations (cont d) The work in the application of CFD techniques to two-phase flows was developed for more than a decade, though nowadays it is still noted that the obtained models are not yet so mature as the ones for single-phase flows (foreword to Nucl.. Eng. Des., 240 (2010)) The field is therefore one of active research, requiring huge computational resources; the brand name of Computational Multi- Fluid Dynamics (CMFD) was proposed for this field of research by Prof. Yadigaroglu (Int. J. Multiph.. Flow, 23, 2003) In principle, DNS, LES and RANS techniques can be all used for two- phase flow,, though the scenario of their application is strongly changed with respect to single-phase In particular, in addition to the integral length scale and the smallest turbulent scale, the scales of two-phase flow structures (e.g., bubbles) are called into play 47

Two-phase flow applications Few general considerations (cont d) In the case of the RANS approach, mass energy and momentum balance equations are written in 3D geometry for each phase k (see e.g., Bestion et al. 2005; Mimouni et al., 2008, Galassi et al., 2009 for NEPTUNE) α k ρk αk ρkwk T + ( α k ρkwk ) = Γk ( αk ρkwk wk ) αk p M k αk ρk g + = + + + αk ( τ k + τ k ) t t 2 2 2 w w k k p w k T α k ρk hk + + α k ρk hk + wk = αk + αk ρk g wk + Γ k hk, i + + q k, iai + q w, k αk ( qk + qk ) t 2 2 t 2 These equations are accompanied by an extension to two-phase flow of a k-ε model k k 1 µ k ρ α ρ ε T k k k k i k + wk, i = k + k [ Production terms k ] + PK t xi αk x j σ K x j ε ε 1 µ ε ε ρ α ρ ε T k k k k k i k + wk, i = k + k [ Cε1Production terms Cε1 k ] + Pε t xi αk x j σ ε x j kk where additional terms of turbulence production appear due to the interaction between the phases. An interfacial area concentration transport equation is also used 48

Two-phase flow applications Few general considerations (cont d) Needless to say, this model relies on the Boussinesq assumption; ; turbulent viscosity is moreover given simply by = C T k µ k µρk ε k Its is quite clear that the success of such a model is strictly linked to its ingredients in terms of constitutive relationships that must be suitable for the particular considered flow regime In particular, for a bubbly flow the momentum transfer term, M k, should account for mass transfer,, the drag and lift forces, the added mass term and the turbulent dispersion of bubbles k 2 A major lack of RANS approaches is anyway in the fact that some two-phase flow fields are naturally unstable: time averaging is therefore suitable only to have a global averaged picture of what happens, loosing instantaneous details (see e.g., the discussion in Yadigaroglu et al., 2008) 49

Two-phase flow applications Few general considerations (cont d) By the way, unsteady calculations with RANS may show oscillations that may somehow match with experimental observations (Zboray( and De Cahard,, 2005) LES models,, of course, reintroduce the possibility to address varying flow fields like the fluctuations of bubble plumes; such applications are interestingly discussed, among the others, by Yadigaroglu et al., (2008) and in works there referred to, and by Niceno et al., (2008) In such discussions, it can be noted that, in similarity with the case of RANS, LES models require accurate closure models for the different terms appearing in the equations in addition to adequate SGS models 50

Two-phase flow applications Few general considerations (cont d) Lahey (2009) recently discussed the capabilities of DNS models in representing two-phase flows As in case of single-phase flow, the attractiveness of this technique lies in the fact that there is no need to introduce empirical models to obtain accurate predictions; the obvious drawback is the heavy computational load In the case of two-phase flows, interface tracking algorithms must be introduced; in the mentioned paper, an algorithm based on the signed distance form the interface is used in the PHASTA code Dam break problems, bubble interactions and plunging jets are within the predictive capabilities, whenever appropriate computational resources are made available 51 CFD-Figure Figure-2.ppt

Prediction of heat transfer deterioration Addressed experimental data As in Sharabi et al. [2007], the considered experimental data are those by Pis menny et al. [2006]: National Technological University of Ukraine turbulent heat transfer in vertical tubes for supercritical water operating pressure of 23.5 MPa inlet temperature and heating conditions involved in these analyses resulted in both dense and gas-like fluid to be present in the test section thin wall stainless steel tubes with inner diameters of 6.28 and 9.50 mm were adopted, with a 600 mm long heated section preceded by a 64 diameters long unheated region cromel-alumel thermocouples were adopted to measure the inlet and outlet fluid temperature, as well as the outer temperature of the tubes. 52

Prediction of heat transfer deterioration Previous results Previous results obtained by Sharabi et al. [2007] with an in-house code a) 6.28 mm ID, q =390 kw/m 2, G= 590 kg/(m 2 s), T inlet =300 C, upward flow b) 6.28 mm ID, q =390 kw/m 2, G= 590 kg/(m 2 s), T inlet =300 C, downward flow (AKN = Abe et al. [1994]; CH = Chien [1982]; JL = Jones and Launder [1972]; LB = Lam and Bremhorst, [1981]; LS = Launder and Sharma [1974]; YS = Yang and Shih [1993], WI=Wilcox [1994], SP=Speziale et al. [1990]) 53

Prediction of heat transfer deterioration Previous results (cont d) It can be noted that: k-ε models predict in a qualitatively reasonable way the onset of heat transfer deterioration occurring in upward flow however, despite of quantitative differences between the results of the different k-ε models, they all tend to predict a larger wall temperature increase than observed on the other hand, the Wilcox [1994] k-ω model (WI) and the Speziale et al. [1990] k-τ model (SP) were seen to predict no deterioration or a very delayed one in the case of upward flow, all the models provided similar results, characterised by the absence of any deterioration phenomenon, in qualitative agreement with experimental observations 54

Prediction of heat transfer deterioration Previous results (cont d) Buoyancy forces accelerate the flow at the wall and lead to an m-shaped velocity profile Reasons for Heat Transfer Deterioration (Longer pipe) Velocity distribution predicted by the YS model (upward flow, G=509 kg/(m2s), q=390 kw/m2, tin=300 C) Velocity distribution predicted by the WI model (upward flow, with G=509 kg/(m2s), q=390 kw/m2, tin=300 C) 55

Prediction of heat transfer deterioration Previous results (cont d) In the transition to the m-shaped profile velocity gradients are suppressed and turbulence production decreases (Longer pipe) Turbulent kinetic energy distribution predicted by the YS model (upward flow, G=509 kg/(m2s), q=390 kw/m2, tin=300 C) Turbulent kinetic energy distribution predicted by the WI model (upward flow, G=509 kg/(m2s), q=390 kw/m2, tin=300 C) 56

Prediction of heat transfer deterioration STAR-CCM+ Results With the STAR-CCM+ code, the following modelling choices were made: The adopted 2D axi-symmetric mesh included 20 radial nodes in a 0.54 mm thick prismatic layer region close to the wall 26 uniform nodes in the remaining core region, having a radius of 2.6 mm The stretching factor adopted in the prismatic layer was 1.2 Trimmed meshes were selected for the core region Though slightly coarser than in the in-house code calculations, the grid was found to be suitable to provide enough accurate results with a reasonable computational effort Later, the results obtained by this grid have been compared to those obtained by a finer one (68 radial and 500 axial nodes) showing little differences Default code options were adopted in relation to advection schemes (2nd order) The steady-state iteration algorithm of the code was adopted, starting with coupled flow and energy iterations and then shifting to the segregated equation approach In all the code runs, it was checked that the requirement y+ < 1 was respected with due margin 57

Prediction of heat transfer deterioration STAR-CCM+ Results (cont d) Concerning water properties at 23.5 MPa, the code allows assigning the dependence of density and specific heat on temperature in polynomial form Thermal conductivity and dynamic viscosity can be instead assigned adopting user defined field functions. Suitable local cubic spline polynomials were then used for these properties, whose coefficients were generated on the basis of tables obtained by the NIST package 1200 200000 0.8 200000 0.50 2.0E-03 Thermal Density Conductivity Cp [J/(kgK)] [kg/m 3 [W/(mK)] ] 180000 0.7 1000 160000 0.6 140000 800 0.5 120000 100000 0.4 600 80000 0.3 400 60000 0.2 40000 200 0.1 20000 Data Splines Interval Boundaries Thermal Dynamic Conductivity Cp Viscosity [J/(kgK)] [kg/(ms)] [W/(mK)] 180000 0.45 1.8E-03 160000 0.40 1.6E-03 140000 0.35 1.4E-03 120000 0.30 1.2E-03 100000 0.25 1.0E-03 0.20 8.0E-04 80000 0.15 6.0E-04 60000 0.10 4.0E-04 40000 0.05 2.0E-04 20000 Data Splines Interval Boundaries 0.0 0 0 0 0640 200 200 645400 650 600 800 655 1000 660 1200 665 1400 6701600 675 1800 2000 680 Temperature [K] [K] 0.0E+00 0.00 0 640 00 200 650 400 600 660 800 670 1000 1200 680 1400 1600 690 1800 2000 700 Temperature [K] [K] 58

Prediction of heat transfer deterioration STAR-CCM+ Results (cont d) The analysis reported herein was limited to four k-ε models: the Two-Layer All y+ Wall Treatment (referred to in the following as all y+ ), suggested for simulating with a reasonable accuracy different kinds of flows; the standard Low-Reynolds Number K-Epsilon Model (referred to in the following as low-re ) suggested by code guidelines for natural convection problems and referred to a model published by Lien et al. [1996]; the AKN model, already used with the in-house code [Abe et al., 1994]; the V2F model that, besides the k and ε equations, solves two additional transport and algebraic equations; this model is suggested to capture more accurately near wall phenomena [Durbin, 1991; Durbin, 1996; Lien et al., 1998]. 59

Prediction of heat transfer deterioration STAR-CCM+ Results (cont d) 900 Wall Temperature [ C] 800 700 600 500 400 Low-Re AKN V2F All y+ Low-Re (finer mesh) Experiment 300 0 20 40 60 80 100 x / D a) 6.28 mm ID, q =390 kw/m 2, G= 590 kg/(m 2 s), T inlet =300 C, upward flow 60

Prediction of heat transfer deterioration STAR-CCM+ Results (cont d) 900 Wall Temperature [ C] 800 700 600 500 400 Low-Re AKN V2F All y+ Experiment 300 0 20 40 60 80 100 x / D a) 6.28 mm ID, q =390 kw/m 2, G= 590 kg/(m 2 s), T inlet =300 C, downward flow 61

Prediction of heat transfer deterioration STAR-CCM+ Results (cont d) It can be noted that: the Two-Layer All y+ Wall Treatment was unable to detect the start of deterioration phenomena in upward flow all the other k-ε models showed a behaviour similar to the one already observed in the previous study: they are able to detect the onset of deterioration they tend to overestimate the effect of deterioration on wall temperature prediction all the models have no difficulty to predict the behaviour observed in downward flow, in which no deterioration was detected The reasons of this behaviour were found to be the same as observed in the previous study (see below) 62

Prediction of heat transfer deterioration STAR-CCM+ Results (cont d) 1.2 1.2 Low-Re Model, Upward Flow AKN Model, Upward Flow 1 1 X-Velocity Component [m/s] 0.8 0.6 0.4 Pipe Inlet 0 x/d 16 32 48 64 80 88 X-Velocity Component [m/s] 0.8 0.6 0.4 Pipe Inlet 0 x/d 16 32 48 64 80 88 0.2 0.2 0 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 Radius [m] Radius [m] 1.2 V2F Model, Upward Flow 1.2 All y+ Model, Upward Flow 1 1 X-Velocity Component [m/s] 0.8 0.6 0.4 Pipe Inlet 0 x/d 16 32 48 64 80 88 X-Velocity Component [m/s] 0.8 0.6 0.4 Pipe Inlet 0 x/d 16 32 48 64 80 88 0.2 0.2 0 0 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 Radius [m] Radius [m] Figure 1: Radial distribution of the axial velocity component in the upward flow case 63

Prediction of heat transfer deterioration STAR-CCM+ Results (cont d) 0.008 0.008 0.007 Low-Re Model, Upward Flow 0.007 AKN Model, Upward Flow Turbulent Kinetic Energy [J/kg] 0.006 0.005 0.004 0.003 0.002 Pipe Inlet 0 x/d 16 32 48 64 80 88 Turbulent Kinetic Energy [J/kg] 0.006 0.005 0.004 0.003 0.002 Pipe Inlet 0 x/d 16 32 48 64 80 88 0.001 0.001 0.000 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.000 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 Radius [m] Radius [m] 0.008 0.008 0.007 V2F Model, Upward Flow 0.007 All y+ Model, Upward Flow Turbulent Kinetic Energy [J/kg] 0.006 0.005 0.004 0.003 0.002 Pipe Inlet 0 x/d 16 32 48 64 80 88 Turbulent Kinetic Energy [J/kg] 0.006 0.005 0.004 0.003 0.002 Pipe Inlet 0 x/d 16 32 48 64 80 88 0.001 0.001 0.000 0.000 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 Radius [m] Radius [m] Figure 1: Radial distribution of turbulent kinetic energy in the upward flow case 64

Prediction of heat transfer deterioration STAR-CCM+ Results (cont d) 1 1 0.9 0.9 X-Velocity Component [m/s] 0.8 0.7 0.6 0.5 0.4 0.3 Pipe Inlet 0 x/d 16 32 48 64 80 88 X-Velocity Component [m/s] 0.8 0.7 0.6 0.5 0.4 0.3 Pipe Inlet 0 x/d 16 32 48 64 80 88 0.2 0.2 0.1 Low-Re Model, Downward Flow 0.1 AKN Model, Downward Flow 0 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 Radius [m] Radius [m] 1 1 0.9 0.9 X-Velocity Component [m/s] 0.8 0.7 0.6 0.5 0.4 0.3 Pipe Inlet 0 x/d 16 32 48 64 80 88 X-Velocity Component [m/s] 0.8 0.7 0.6 0.5 0.4 0.3 Pipe Inlet 0 x/d 16 32 48 64 80 88 0.2 0.2 0.1 V2F Model, Downward Flow 0.1 All y+ Model, Downward Flow 0 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 Radius [m] 0 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 Radius [m] Figure 1: Radial distribution of the axial velocity component in the downward flow case 65

Prediction of heat transfer deterioration STAR-CCM+ Results (cont d) 0.008 0.008 0.007 Low-Re Model, Downward Flow 0.007 AKN Model, Downward Flow Turbulent Kinetic Energy [J/kg] 0.006 0.005 0.004 0.003 0.002 Pipe Inlet 0 x/d 16 32 48 64 80 88 Turbulent Kinetic Energy [J/kg] 0.006 0.005 0.004 0.003 0.002 Pipe Inlet 0 x/d 16 32 48 64 80 88 0.001 0.001 0.000 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.000 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 Radius [m] Radius [m] 0.008 0.008 0.007 V2F Model, Downward Flow 0.007 All y+ Model, Downward Flow Turbulent Kinetic Energy [J/kg] 0.006 0.005 0.004 0.003 0.002 Pipe Inlet 0 x/d 16 32 48 64 80 88 Turbulent Kinetic Energy [J/kg] 0.006 0.005 0.004 0.003 0.002 Pipe Inlet 0 x/d 16 32 48 64 80 88 0.001 0.001 0.000 0.000 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 Radius [m] Radius [m] Figure 1: Radial distribution of turbulent kinetic energy in the downward flow case 66

In summary CFD and CMFD are very powerful tools, whose capabilities are conditioned to our understanding of phenomena and to computer power The smaller is the degree of empiricism we wish to introduce in the models, the greatest is the computer power needed It is a very fascinating world in which smart ideas are needed to discover newer and newer possibilities 67

Thank you for your attention, Walter Ambrosini 68

Sources and suggested readings N.E. Todreas,, M. S. Kazimi Nuclear Systems I, I, Taylor & Francis, 1990. D.J. Tritton Physical Fluid Dynamics,, Oxford Science Publications, 2 nd Edition, 1997. H.K. Veersteg and W. Malalasekera An introduction to computational fluid dynamics,, Pearson, Prentice Hall, 1995. D.C. Wilcox Turbulence Modeling for CFD,, 2nd Edition, DCW Industries, 1998. E. Baglietto,, H. Ninokata, Takeharu Misawa, CFD and DNS methodologies development for fuel bundle simulations, s, Nuclear Engineering and Design 236 (2006) 1503 1510 1510 Maria Cristina Galassi,, Pierre Coste,, Christophe Morel and Fabio Moretti,, Two-Phase Flow Simulations for PTS Investigation by Means of Neptune CFD Code, Hindawi Publishing Corporation, Science and Technology of Nuclear Installations, Volume 2009, Article ID 950536, 12 pages, doi:10.1155/2009/950536 1155/2009/950536 D. Bestion and A. Guelfi,, Status and Perspective of Two-Phase Flow Modelling in the Neptune Multiscale Thiermal- Hydraulic Platform for Nuclear Reactor Simulation, NUCLEAR ENGINEERING EERING AND TECHNOLOGY, VOL.37 NO.6 DECEMBER 2005 S. Mimouni,, M. Boucker J. Laviéville ville,, A. Guelfi,, D. Bestion, Modelling and computation of cavitation and boiling bubbly flows with the NEPTUNE CFD code, Nuclear Engineering and Design 238 (2008) 680 692 692 G. Yadigaroglu,, M. Simiano,, R. Milenkovic,, J. Kubasch M. Milelli,, R. Zboray,, F. De Cachard,, B. Smith,, D. Lakehal,, B. Sigg, CFD4NRS with a focus on experimental and CMFD investigations of bubbly flows, Nuclear Engineering and Design 238 (2008) 771 785 785 Richard T. Lahey Jr., On the direct numerical simulation of two-phase flows, Nuclear Engineering and Design 239 (2009) 867 879 879 R. Zboray,, F. de Cachard, Simulating large-scale bubble plumes using various closure and two-phase turbulence models, Nuclear Engineering and Design 235 (2005) 867 884 884 B. Niceno,, M.T. Dhotre,, N.G. Deen,, One-equation equation sub-grid scale (SGS) modelling for Euler-- --Euler large eddy simulation (EELES) of dispersed bubbly flow, Chemical Engineering Science 636 3 (2008) 3923 3931 Walter Ambrosini, Continuing Assessment of System and CFD Codes for Heat Transfer and Stability in Supercritical Fluids, 4th International Symposium on Supercritical Water-Cooled Reactors, March 8-11, 8 2009, Heidelberg, Germany, Paper No. 83 SCIENTECH Inc., 1999, RELAP5/Mod3 Code Manual, Volume I: Code Structure, System Models and Solution Methods,, The Thermal Hydraulics Group, Idaho, June 1999. 69